Chapter 12 Algebraic Expressions

The solution is presented in a table format below. For each expression, we identify the terms that contain variables (i.e., are not constants) and then state their corresponding numerical coefficients. The numerical coefficient is the number part of a term that is multiplied by the variable(s).

Expression	Term (which is not a constant)	Numerical Coefficient
$xy + 4$	$xy$	1
$13 – y^2$	$-y^2$	-1
$13 – y + 5y^2$	$-y$	-1
	$5y^2$	5
$4p^2q – 3pq^2 + 5$	$4p^2q$	4
	$-3pq^2$	-3

(a) Coefficients of x

The coefficient of x in a term is the remaining part of the term that multiplies x.

Expression	Term containing x	Coefficient of x
$4x \ - \ 3y$	$4x$	4
$8 \ - \ x \ + \ y$	$-x$	-1
$y^2x \ - \ y$	$y^2x$	$y^2$
$2z \ - \ 5xz$	$-5xz$	$-5z$

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(b) Coefficients of y

The coefficient of y in a term is the remaining part of the term that multiplies y.

Expression	Term containing y	Coefficient of y
$4x \ - \ 3y$	$-3y$	-3
$8 \ + \ yz$	$yz$	$z$
$yz^2 \ + \ 5$	$yz^2$	$z^2$
$my \ + \ m$	$my$	$m$

Definition: Like terms are terms that have the same algebraic factors (variables raised to the same powers). Unlike terms have different algebraic factors.

S.No.	Pair of Terms	Algebraic Factors	Like/Unlike Terms	Reason
(i)	$7x , 12y$	$x$ ; $y$	Unlike	The terms have different algebraic factors (variables).
(ii)	$15x , -21x$	$x$ ; $x$	Like	The terms have the same algebraic factor ($x$).
(iii)	$-4ab , 7ba$	$a \times b$ ; $b \times a$	Like	The terms have the same algebraic factors ($a$ and $b$). Note that $ba$ is the same as $ab$.
(iv)	$3xy , 3x$	$x \times y$ ; $x$	Unlike	The algebraic factors are different; one term has factor $y$, the other does not.
(v)	$6xy^2 , 9x^2y$	$x \times y \times y$ ; $x \times x \times y$	Unlike	The powers of the variables are different ($y^2$ vs $y$, $x$ vs $x^2$).
(vi)	$pq^2 , -4pq^2$	$p \times q \times q$ ; $p \times q \times q$	Like	The terms have the same algebraic factors ($p$ and $q^2$).
(vii)	$mn^2 , 10mn$	$m \times n \times n$ ; $m \times n$	Unlike	The powers of the variable $n$ are different ($n^2$ vs $n$).

(i) Subtraction of z from y

The expression is $y - z$.

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(ii) One-half of the sum of numbers x and y

The sum of numbers x and y is $(x + y)$.

One-half of the sum is $\frac{1}{2}(x + y)$ or $\frac{x+y}{2}$.

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(iii) The number z multiplied by itself

The expression is $z \times z = z^2$.

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(iv) One-fourth of the product of numbers p and q

The product of numbers p and q is $(p \times q)$ or $pq$.

One-fourth of the product is $\frac{1}{4}(pq)$ or $\frac{pq}{4}$.

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(v) Numbers x and y both squared and added

x squared is $x^2$.

y squared is $y^2$.

The expression is $x^2 + y^2$.

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(vi) Number 5 added to three times the product of numbers m and n

The product of numbers m and n is $mn$.

Three times the product is $3(mn)$ or $3mn$.

Number 5 added to this is $3mn + 5$.

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(vii) Product of numbers y and z subtracted from 10

The product of numbers y and z is $yz$.

Subtracting the product from 10 gives $10 - yz$.

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(viii) Sum of numbers a and b subtracted from their product

The sum of numbers a and b is $(a + b)$.

The product of numbers a and b is $ab$.

Subtracting the sum from the product gives $ab - (a + b)$.

(i) Identification of Terms and Factors by Tree Diagrams

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(a) $x – 3$

Expression: $x-3$

Terms: $x$, $-3$

Factors: $(x)$, $(-3)$

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(b) $1 + x + x^2$

Expression: $1 + x + x^2$

Terms: $1$, $x$, $x^2$

Factors: $(1)$, $(x)$, $(x, x)$

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(c) $y – y^3$

Expression: $y – y^3$

Terms: $y$, $-y^3$

Factors: $(y)$, $(-1, y, y, y)$

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(d) $5xy^2 + 7x^2y$

Expression: $5xy^2 + 7x^2y$

Terms: $5xy^2$, $7x^2y$

Factors: $(5, x, y, y)$, $(7, x, x, y)$

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(e) $– ab + 2b^2 – 3a^2$

Expression: $– ab + 2b^2 – 3a^2$

Terms: $-ab$, $2b^2$, $-3a^2$

Factors: $(-1, a, b)$, $(2, b, b)$, $(-1, 3, a, a)$

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(ii) Identification of Terms and Factors in a Table

Expression	Terms	Factors
(a) $– 4x + 5$	$-4x$	$-4, x$
	$5$	$5$
(b) $– 4x + 5y$	$-4x$	$-4, x$
	$5y$	$5, y$
(c) $5y + 3y^2$	$5y$	$5, y$
	$3y^2$	$3, y, y$
(d) $xy + 2x^2y^2$	$xy$	$x, y$
	$2x^2y^2$	$2, x, x, y, y$
(e) $pq + q$	$pq$	$p, q$
	$q$	$q$
(f) $1.2 ab – 2.4 b + 3.6 a$	$1.2ab$	$1.2, a, b$
	$-2.4b$	$-2.4, b$
	$3.6a$	$3.6, a$
(g) $\frac{3}{4}x + \frac{1}{4}$	$\frac{3}{4}x$	$\frac{3}{4}, x$
	$\frac{1}{4}$	$\frac{1}{4}$
(h) $0.1 p^2 + 0.2 q^2$	$0.1p^2$	$0.1, p, p$
	$0.2q^2$	$0.2, q, q$

The following table lists the given expressions, identifies the terms which are not constants (i.e., terms containing variables), and provides their corresponding numerical coefficients.

Expression	Term (other than constant)	Numerical Coefficient
(i) $5 – 3t^2$	$-3t^2$	-3
(ii) $1 + t + t^2 + t^3$	$t$	1
	$t^2$	1
	$t^3$	1
(iii) $x + 2xy + 3y$	$x$	1
	$2xy$	2
	$3y$	3
(iv) $100m + 1000n$	$100m$	100
	$1000n$	1000
(v) $–p^2q^2 + 7pq$	$-p^2q^2$	-1
	$7pq$	7
(vi) $1.2 a + 0.8 b$	$1.2a$	1.2
	$0.8b$	0.8
(vii) $3.14 r^2$	$3.14r^2$	3.14
(viii) $2 (l + b) = 2l + 2b$	$2l$	2
	$2b$	2
(ix) $0.1 y + 0.01 y^2$	$0.1y$	0.1
	$0.01y^2$	0.01

(a) Identify terms which contain x and give the coefficient of x.

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The coefficient of a specific variable in a term is found by taking all other parts of the term (including the numerical coefficient and other variables) that are multiplied with it.

Expression	Term containing 'x'	Coefficient of 'x'
(i) $y^2x + y$	$y^2x$	$y^2$
(ii) $13y^2 – 8yx$	$-8yx$	$-8y$
(iii) $x + y + 2$	$x$	1
(iv) $5 + z + zx$	$zx$	$z$
(v) $1 + x + xy$	$x$	1
	$xy$	$y$
(vi) $12xy^2 + 25$	$12xy^2$	$12y^2$
(vii) $7x + xy^2$	$7x$	7
	$xy^2$	$y^2$

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(b) Identify terms which contain y² and give the coefficient of y².

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Similar to the above, the coefficient of $y^2$ will be the remaining factors in the term containing $y^2$.

Expression	Term containing 'y2'	Coefficient of 'y2'
(i) $8 – xy^2$	$-xy^2$	$-x$
(ii) $5y^2 + 7x$	$5y^2$	5
(iii) $2x^2y – 15xy^2 + 7y^2$	$-15xy^2$	$-15x$
	$7y^2$	7

We classify algebraic expressions based on the number of unlike terms they contain:

A Monomial is an expression that contains only one term.

A Binomial is an expression that contains two unlike terms.

A Trinomial is an expression that contains three unlike terms.

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Based on these definitions, the classification of the given expressions is as follows:

Expression	Number of Terms	Classification
(i) $4y – 7z$	2	Binomial
(ii) $y^2$	1	Monomial
(iii) $x + y – xy$	3	Trinomial
(iv) $100$	1	Monomial
(v) $ab – a – b$	3	Trinomial
(vi) $5 – 3t$	2	Binomial
(vii) $4p^2q – 4pq^2$	2	Binomial
(viii) $7mn$	1	Monomial
(ix) $z^2 – 3z + 8$	3	Trinomial
(x) $a^2 + b^2$	2	Binomial
(xi) $z^2 + z$	2	Binomial
(xii) $1 + x + x^2$	3	Trinomial

Two or more terms are considered like terms if they have the same algebraic factors. This means they must have the same variables, and each variable must have the same exponent. The numerical coefficients of like terms can be different.

Terms that do not have the same algebraic factors are called unlike terms.

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The classification of the given pairs of terms is as follows:

Pair of Terms	Algebraic Factors	Classification (Like/Unlike)
(i) $1, 100$	Both are constants with no variable factors.	Like Terms
(ii) $–7x, \frac{5}{2}x$	Both terms have the same algebraic factor: $x$.	Like Terms
(iii) $– 29x, – 29y$	The first term has the factor $x$, and the second has the factor $y$. The factors are different.	Unlike Terms
(iv) $14xy, 42yx$	Both terms have the same algebraic factors: $x$ and $y$ (since $xy = yx$).	Like Terms
(v) $4m^2p, 4mp^2$	The factors are $m^2, p$ and $m, p^2$. The exponents of the variables are different.	Unlike Terms
(vi) $12xz, 12x^2z^2$	The factors are $x, z$ and $x^2, z^2$. The exponents of the variables are different.	Unlike Terms

Like terms are terms that have the same variables raised to the same powers. The coefficients of the like terms can be different. We will group the like terms from the given lists.

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(a) Like terms in the list:

– xy² , – 4yx² , 8x² , 2xy² , 7y , – 11x² , –100x , –11yx , 20x²y , – 6x² , y , 2xy , 3x

The groups of like terms are:

Group 1 (with factors $x, y^2$): $– xy^2$, $2xy^2$

Group 2 (with factors $x^2, y$): $– 4yx^2$, $20x^2y$

Group 3 (with factor $x^2$): $8x^2$, $– 11x^2$, $– 6x^2$

Group 4 (with factor $y$): $7y$, $y$

Group 5 (with factor $x$): $– 100x$, $3x$

Group 6 (with factors $x, y$): $– 11yx$, $2xy$

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(b) Like terms in the list:

10pq , 7p , 8q , – p²q² , – 7qp , – 100q , – 23 , 12q²p² , – 5p² , 41 , 2405p , 78qp , 13p²q , qp² , 701p²

The groups of like terms are:

Group 1 (with factors $p, q$): $10pq$, $– 7qp$, $78qp$

Group 2 (with factor $p$): $7p$, $2405p$

Group 3 (with factor $q$): $8q$, $– 100q$

Group 4 (with factors $p^2, q^2$): $– p^2q^2$, $12q^2p^2$

Group 5 (constants): $– 23$, $41$

Group 6 (with factor $p^2$): $– 5p^2$, $701p^2$

Group 7 (with factors $p^2, q$): $13p^2q$, $qp^2$

Given

The expression to simplify is $12m^2 – 9m + 5m – 4m^2 – 7m + 10$.

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Solution

First, we identify the like terms in the expression.

Terms with $m^2$: $12m^2$ and $-4m^2$.

Terms with $m$: $-9m$, $5m$, and $-7m$.

Constant term: $10$.

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Next, we rearrange the expression to group the like terms together:

$12m^2 – 4m^2 – 9m + 5m – 7m + 10$

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Now, we combine the coefficients of the like terms:

Combining $m^2$ terms: $(12 - 4)m^2 = 8m^2$.

Combining $m$ terms: $(-9 + 5 - 7)m = (-4 - 7)m = -11m$.

The constant term remains $10$.

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Finally, we write the simplified expression:

$8m^2 - 11m + 10$

Therefore, the simplified expression is $8m^2 - 11m + 10$.

Given

We need to subtract the expression $(24ab – 10b – 18a)$ from the expression $(30ab + 12b + 14a)$.

This can be written as: $(30ab + 12b + 14a) - (24ab – 10b – 18a)$.

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Solution (Method 1: Row Method)

To subtract, we remove the parentheses. Remember that subtracting an expression is equivalent to adding its additive inverse. So, we change the sign of each term in the expression being subtracted.

$(30ab + 12b + 14a) - (24ab – 10b – 18a)$

$= 30ab + 12b + 14a - 24ab - (-10b) - (-18a)$

$= 30ab + 12b + 14a - 24ab + 10b + 18a$

Now, we group the like terms together:

$= (30ab - 24ab) + (12b + 10b) + (14a + 18a)$

Combine the coefficients of the like terms:

$= (30 - 24)ab + (12 + 10)b + (14 + 18)a$

$= 6ab + 22b + 32a$

Therefore, the result of the subtraction is $6ab + 22b + 32a$.

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Alternate Solution (Method 2: Column Method)

We arrange the expressions vertically, aligning the like terms. Then, we change the sign of each term in the expression being subtracted and add the expressions.

$\begin{array}{r} & 30ab & + & 12b & + & 14a \\ - & (24ab & - & 10b & - & 18a) \\ \hline \end{array}$

Changing the signs of the second expression and adding:

$\begin{array}{r} & 30ab & + & 12b & + & 14a \\ & -24ab & + & 10b & + & 18a \\ \hline & 6ab & + & 22b & + & 32a \\ \hline \end{array}$

The result is $6ab + 22b + 32a$.

Given

First set of expressions: $(2y^2 + 3yz)$, $(-y^2 - yz - z^2)$, $(yz + 2z^2)$.

Second set of expressions: $(3y^2 - z^2)$, $(-y^2 + yz + z^2)$.

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To Find

We need to find the result of subtracting the sum of the second set of expressions from the sum of the first set of expressions.

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Solution

Step 1: Find the sum of the first three expressions.

Let Sum 1 be the sum of the first three expressions:

Sum 1 = $(2y^2 + 3yz) + (-y^2 - yz - z^2) + (yz + 2z^2)$

Removing parentheses:

$= 2y^2 + 3yz - y^2 - yz - z^2 + yz + 2z^2$

Grouping like terms:

$= (2y^2 - y^2) + (3yz - yz + yz) + (-z^2 + 2z^2)$

Combining like terms:

$= (2-1)y^2 + (3-1+1)yz + (-1+2)z^2$

$= y^2 + 3yz + z^2$

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Step 2: Find the sum of the next two expressions.

Let Sum 2 be the sum of the next two expressions:

Sum 2 = $(3y^2 - z^2) + (-y^2 + yz + z^2)$

Removing parentheses:

$= 3y^2 - z^2 - y^2 + yz + z^2$

Grouping like terms:

$= (3y^2 - y^2) + (yz) + (-z^2 + z^2)$

Combining like terms:

$= (3-1)y^2 + yz + (-1+1)z^2$

$= 2y^2 + yz + 0z^2$

$= 2y^2 + yz$

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Step 3: Subtract the second sum (Sum 2) from the first sum (Sum 1).

Result = (Sum 1) - (Sum 2)

$= (y^2 + 3yz + z^2) - (2y^2 + yz)$

Removing parentheses (change the sign of each term in the expression being subtracted):

$= y^2 + 3yz + z^2 - 2y^2 - yz$

Grouping like terms:

$= (y^2 - 2y^2) + (3yz - yz) + z^2$

Combining like terms:

$= (1 - 2)y^2 + (3 - 1)yz + z^2$

$= -y^2 + 2yz + z^2$

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Therefore, the final result is $-y^2 + 2yz + z^2$.

(i) $21b – 32 + 7b – 20b$

Group the like terms:

$= (21b + 7b - 20b) - 32$

Combine the coefficients of $b$:

$= (21 + 7 - 20)b - 32$

$= (28 - 20)b - 32$

$= 8b - 32$

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(ii) $– z^2 + 13z^2 – 5z + 7z^3 – 15z$

Group the like terms (usually in descending order of power):

$= 7z^3 + (-z^2 + 13z^2) + (-5z - 15z)$

Combine the coefficients:

$= 7z^3 + (-1 + 13)z^2 + (-5 - 15)z$

$= 7z^3 + 12z^2 - 20z$

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(iii) $p – (p – q) – q – (q – p)$

Distribute the negative signs:

$= p - p + q - q - q + p$

Group the like terms:

$= (p - p + p) + (q - q - q)$

Combine the coefficients:

$= (1 - 1 + 1)p + (1 - 1 - 1)q$

$= 1p - 1q$

$= p - q$

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(iv) $3a – 2b – ab – (a – b + ab) + 3ab + b – a$

Distribute the negative sign:

$= 3a - 2b - ab - a + b - ab + 3ab + b - a$

Group the like terms:

$= (3a - a - a) + (-2b + b + b) + (-ab - ab + 3ab)$

Combine the coefficients:

$= (3 - 1 - 1)a + (-2 + 1 + 1)b + (-1 - 1 + 3)ab$

$= (1)a + (0)b + (1)ab$

$= a + ab$

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(v) $5x^2y – 5x^2 + 3yx^2 – 3y^2 + x^2 – y^2 + 8xy^2 – 3y^2$

Recognize that $yx^2$ is the same as $x^2y$. Group the like terms:

$= (5x^2y + 3yx^2) + (-5x^2 + x^2) + (-3y^2 - y^2 - 3y^2) + 8xy^2$

Combine the coefficients:

$= (5 + 3)x^2y + (-5 + 1)x^2 + (-3 - 1 - 3)y^2 + 8xy^2$

$= 8x^2y - 4x^2 - 7y^2 + 8xy^2$

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(vi) $(3y^2 + 5y – 4) – (8y – y^2 – 4)$

Distribute the negative sign:

$= 3y^2 + 5y - 4 - 8y - (-y^2) - (-4)$

$= 3y^2 + 5y - 4 - 8y + y^2 + 4$

Group the like terms:

$= (3y^2 + y^2) + (5y - 8y) + (-4 + 4)$

Combine the coefficients:

$= (3 + 1)y^2 + (5 - 8)y + 0$

$= 4y^2 - 3y$

(i) Add: $3mn, – 5mn, 8mn, – 4mn$

Sum = $3mn + (-5mn) + 8mn + (-4mn)$

$= 3mn - 5mn + 8mn - 4mn$

Group the terms (all are like terms):

$= (3 - 5 + 8 - 4)mn$

$= (11 - 9)mn$

$= 2mn$

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(ii) Add: $t – 8tz, 3tz – z, z – t$

Sum = $(t - 8tz) + (3tz - z) + (z - t)$

$= t - 8tz + 3tz - z + z - t$

Group like terms:

$= (t - t) + (-8tz + 3tz) + (-z + z)$

Combine coefficients:

$= (1 - 1)t + (-8 + 3)tz + (-1 + 1)z$

$= 0t - 5tz + 0z$

$= -5tz$

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(iii) Add: $–7mn + 5, 12mn + 2, 9mn – 8, – 2mn – 3$

Sum = $(-7mn + 5) + (12mn + 2) + (9mn - 8) + (-2mn - 3)$

$= -7mn + 5 + 12mn + 2 + 9mn - 8 - 2mn - 3$

Group like terms:

$= (-7mn + 12mn + 9mn - 2mn) + (5 + 2 - 8 - 3)$

Combine coefficients:

$= (-7 + 12 + 9 - 2)mn + (7 - 11)$

$= (21 - 9)mn + (-4)$

$= 12mn - 4$

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(iv) Add: $a + b – 3, b – a + 3, a – b + 3$

Sum = $(a + b - 3) + (b - a + 3) + (a - b + 3)$

$= a + b - 3 + b - a + 3 + a - b + 3$

Group like terms:

$= (a - a + a) + (b + b - b) + (-3 + 3 + 3)$

Combine coefficients:

$= (1 - 1 + 1)a + (1 + 1 - 1)b + (0 + 3)$

$= 1a + 1b + 3$

$= a + b + 3$

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(v) Add: $14x + 10y – 12xy – 13, 18 – 7x – 10y + 8xy, 4xy$

Sum = $(14x + 10y - 12xy - 13) + (18 - 7x - 10y + 8xy) + (4xy)$

$= 14x + 10y - 12xy - 13 + 18 - 7x - 10y + 8xy + 4xy$

Group like terms:

$= (14x - 7x) + (10y - 10y) + (-12xy + 8xy + 4xy) + (-13 + 18)$

Combine coefficients:

$= (14 - 7)x + (10 - 10)y + (-12 + 8 + 4)xy + (5)$

$= 7x + 0y + (-12 + 12)xy + 5$

$= 7x + 0xy + 5$

$= 7x + 5$

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(vi) Add: $5m – 7n, 3n – 4m + 2, 2m – 3mn – 5$

Sum = $(5m - 7n) + (3n - 4m + 2) + (2m - 3mn - 5)$

$= 5m - 7n + 3n - 4m + 2 + 2m - 3mn - 5$

Group like terms:

$= (5m - 4m + 2m) + (-7n + 3n) - 3mn + (2 - 5)$

Combine coefficients:

$= (5 - 4 + 2)m + (-7 + 3)n - 3mn + (-3)$

$= (1 + 2)m + (-4)n - 3mn - 3$

$= 3m - 4n - 3mn - 3$

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(vii) Add: $4x^2y, – 3xy^2, –5xy^2, 5x^2y$

Sum = $4x^2y + (-3xy^2) + (-5xy^2) + 5x^2y$

$= 4x^2y - 3xy^2 - 5xy^2 + 5x^2y$

Group like terms:

$= (4x^2y + 5x^2y) + (-3xy^2 - 5xy^2)$

Combine coefficients:

$= (4 + 5)x^2y + (-3 - 5)xy^2$

$= 9x^2y - 8xy^2$

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(viii) Add: $3p^2q^2 – 4pq + 5, – 10 p^2q^2, 15 + 9pq + 7p^2q^2$

Sum = $(3p^2q^2 - 4pq + 5) + (-10p^2q^2) + (15 + 9pq + 7p^2q^2)$

$= 3p^2q^2 - 4pq + 5 - 10p^2q^2 + 15 + 9pq + 7p^2q^2$

Group like terms:

$= (3p^2q^2 - 10p^2q^2 + 7p^2q^2) + (-4pq + 9pq) + (5 + 15)$

Combine coefficients:

$= (3 - 10 + 7)p^2q^2 + (-4 + 9)pq + 20$

$= (10 - 10)p^2q^2 + 5pq + 20$

$= 0p^2q^2 + 5pq + 20$

$= 5pq + 20$

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(ix) Add: $ab – 4a, 4b – ab, 4a – 4b$

Sum = $(ab - 4a) + (4b - ab) + (4a - 4b)$

$= ab - 4a + 4b - ab + 4a - 4b$

Group like terms:

$= (ab - ab) + (-4a + 4a) + (4b - 4b)$

Combine coefficients:

$= (1 - 1)ab + (-4 + 4)a + (4 - 4)b$

$= 0ab + 0a + 0b$

$= 0$

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(x) Add: $x^2 – y^2 – 1, y^2 – 1 – x^2, 1 – x^2 – y^2$

Sum = $(x^2 - y^2 - 1) + (y^2 - 1 - x^2) + (1 - x^2 - y^2)$

$= x^2 - y^2 - 1 + y^2 - 1 - x^2 + 1 - x^2 - y^2$

Group like terms:

$= (x^2 - x^2 - x^2) + (-y^2 + y^2 - y^2) + (-1 - 1 + 1)$

Combine coefficients:

$= (1 - 1 - 1)x^2 + (-1 + 1 - 1)y^2 + (-2 + 1)$

$= -1x^2 - 1y^2 - 1$

$= -x^2 - y^2 - 1$

(i) Subtract –5y² from y²

Solution:

We need to calculate: $y^2 - (-5y^2)$

$= y^2 + 5y^2$

Combine like terms:

$= (1 + 5)y^2$

$= 6y^2$

The result is $6y^2$.

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(ii) Subtract 6xy from –12xy

Solution:

We need to calculate: $-12xy - (6xy)$

$= -12xy - 6xy$

Combine like terms:

$= (-12 - 6)xy$

$= -18xy$

The result is $-18xy$.

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(iii) Subtract (a – b) from (a + b)

Solution:

We need to calculate: $(a + b) - (a - b)$

Remove parentheses, changing signs of the second expression:

$= a + b - a - (-b)$

$= a + b - a + b$

Group like terms:

$= (a - a) + (b + b)$

Combine coefficients:

$= (1 - 1)a + (1 + 1)b$

$= 0a + 2b$

$= 2b$

The result is $2b$.

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(iv) Subtract a(b – 5) from b(5 – a)

Solution:

First, expand the expressions:

$a(b - 5) = ab - 5a$

$b(5 - a) = 5b - ab$

Now, subtract $(ab - 5a)$ from $(5b - ab)$:

$(5b - ab) - (ab - 5a)$

Remove parentheses, changing signs:

$= 5b - ab - ab - (-5a)$

$= 5b - ab - ab + 5a$

Group like terms:

$= 5a + 5b + (-ab - ab)$

Combine coefficients:

$= 5a + 5b + (-1 - 1)ab$

$= 5a + 5b - 2ab$

The result is $5a + 5b - 2ab$.

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(v) Subtract – m² + 5mn from 4m² – 3mn + 8

Solution (Row Method):

$(4m^2 - 3mn + 8) - (-m^2 + 5mn)$

Remove parentheses, changing signs:

$= 4m^2 - 3mn + 8 - (-m^2) - (5mn)$

$= 4m^2 - 3mn + 8 + m^2 - 5mn$

Group like terms:

$= (4m^2 + m^2) + (-3mn - 5mn) + 8$

Combine coefficients:

$= (4 + 1)m^2 + (-3 - 5)mn + 8$

$= 5m^2 - 8mn + 8$

Alternate Solution (Column Method):

Align like terms. Change signs of the expression to be subtracted and add.

$\begin{array}{r} & 4m^2 & - & 3mn & + & 8 \\ - & (-m^2 & + & 5mn & & ) \\ \hline \end{array}$

Change signs and add:

$\begin{array}{r} & 4m^2 & - & 3mn & + & 8 \\ & +m^2 & - & 5mn & & \\ \hline & 5m^2 & - & 8mn & + & 8 \\ \hline \end{array}$

The result is $5m^2 - 8mn + 8$.

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(vi) Subtract – x² + 10x – 5 from 5x – 10

Solution (Row Method):

$(5x - 10) - (-x^2 + 10x - 5)$

Remove parentheses, changing signs:

$= 5x - 10 - (-x^2) - (10x) - (-5)$

$= 5x - 10 + x^2 - 10x + 5$

Group like terms (in descending order of power):

$= x^2 + (5x - 10x) + (-10 + 5)$

Combine coefficients:

$= x^2 + (5 - 10)x + (-5)$

$= x^2 - 5x - 5$

Alternate Solution (Column Method):

Align like terms (use 0 for missing terms). Change signs of the expression to be subtracted and add.

$\begin{array}{r} & & & 5x & - & 10 \\ - & (-x^2 & + & 10x & - & 5) \\ \hline \end{array}$

Change signs and add:

$\begin{array}{r} & & & 5x & - & 10 \\ & +x^2 & - & 10x & + & 5 \\ \hline & x^2 & - & 5x & - & 5 \\ \hline \end{array}$

The result is $x^2 - 5x - 5$.

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(vii) Subtract 5a² – 7ab + 5b² from 3ab – 2a² – 2b²

Solution (Row Method):

$(3ab - 2a^2 - 2b^2) - (5a^2 - 7ab + 5b^2)$

Remove parentheses, changing signs:

$= 3ab - 2a^2 - 2b^2 - 5a^2 - (-7ab) - (5b^2)$

$= 3ab - 2a^2 - 2b^2 - 5a^2 + 7ab - 5b^2$

Group like terms (e.g., by $a^2$, then $ab$, then $b^2$):

$= (-2a^2 - 5a^2) + (3ab + 7ab) + (-2b^2 - 5b^2)$

Combine coefficients:

$= (-2 - 5)a^2 + (3 + 7)ab + (-2 - 5)b^2$

$= -7a^2 + 10ab - 7b^2$

Alternate Solution (Column Method):

Align like terms. Change signs of the expression to be subtracted and add.

$\begin{array}{r} & -2a^2 & + & 3ab & - & 2b^2 \\ - & (5a^2 & - & 7ab & + & 5b^2) \\ \hline \end{array}$

Change signs and add:

$\begin{array}{r} & -2a^2 & + & 3ab & - & 2b^2 \\ & -5a^2 & + & 7ab & - & 5b^2 \\ \hline & -7a^2 & + & 10ab & - & 7b^2 \\ \hline \end{array}$

The result is $-7a^2 + 10ab - 7b^2$.

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(viii) Subtract 4pq – 5q² – 3p² from 5p² + 3q² – pq

Solution (Row Method):

$(5p^2 + 3q^2 - pq) - (4pq - 5q^2 - 3p^2)$

Remove parentheses, changing signs:

$= 5p^2 + 3q^2 - pq - 4pq - (-5q^2) - (-3p^2)$

$= 5p^2 + 3q^2 - pq - 4pq + 5q^2 + 3p^2$

Group like terms (e.g., by $p^2$, then $q^2$, then $pq$):

$= (5p^2 + 3p^2) + (3q^2 + 5q^2) + (-pq - 4pq)$

Combine coefficients:

$= (5 + 3)p^2 + (3 + 5)q^2 + (-1 - 4)pq$

$= 8p^2 + 8q^2 - 5pq$

Alternate Solution (Column Method):

Align like terms. Change signs of the expression to be subtracted and add.

$\begin{array}{r} & 5p^2 & + & 3q^2 & - & pq \\ - & (-3p^2 & - & 5q^2 & + & 4pq) \\ \hline \end{array}$

Change signs and add:

$\begin{array}{r} & 5p^2 & + & 3q^2 & - & pq \\ & +3p^2 & + & 5q^2 & - & 4pq \\ \hline & 8p^2 & + & 8q^2 & - & 5pq \\ \hline \end{array}$

The result is $8p^2 + 8q^2 - 5pq$.

(a) What should be added to $x^2 + xy + y^2$ to obtain $2x^2 + 3xy$?

Given

Initial expression: $x^2 + xy + y^2$

Target expression: $2x^2 + 3xy$

To Find

The expression that should be added to the initial expression to get the target expression.

Solution

Let the required expression be $P$.

According to the problem, $(x^2 + xy + y^2) + P = 2x^2 + 3xy$.

To find $P$, we subtract $(x^2 + xy + y^2)$ from $(2x^2 + 3xy)$.

$P = (2x^2 + 3xy) - (x^2 + xy + y^2)$

Remove the parentheses, changing the sign of each term being subtracted:

$P = 2x^2 + 3xy - x^2 - xy - y^2$

Group the like terms:

$P = (2x^2 - x^2) + (3xy - xy) - y^2$

Combine the coefficients:

$P = (2 - 1)x^2 + (3 - 1)xy - y^2$

$P = x^2 + 2xy - y^2$

Therefore, $x^2 + 2xy - y^2$ should be added to $x^2 + xy + y^2$ to obtain $2x^2 + 3xy$.

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(b) What should be subtracted from $2a + 8b + 10$ to get $– 3a + 7b + 16$?

Given

Initial expression: $2a + 8b + 10$

Resulting expression: $-3a + 7b + 16$

To Find

The expression that should be subtracted from the initial expression to get the resulting expression.

Solution

Let the required expression be $Q$.

According to the problem, $(2a + 8b + 10) - Q = -3a + 7b + 16$.

To find $Q$, we subtract the resulting expression $(-3a + 7b + 16)$ from the initial expression $(2a + 8b + 10)$.

$Q = (2a + 8b + 10) - (-3a + 7b + 16)$

Remove the parentheses, changing the sign of each term being subtracted:

$Q = 2a + 8b + 10 - (-3a) - (7b) - (16)$

$Q = 2a + 8b + 10 + 3a - 7b - 16$

Group the like terms:

$Q = (2a + 3a) + (8b - 7b) + (10 - 16)$

Combine the coefficients:

$Q = (2 + 3)a + (8 - 7)b + (-6)$

$Q = 5a + 1b - 6$

$Q = 5a + b - 6$

Therefore, $5a + b - 6$ should be subtracted from $2a + 8b + 10$ to get $-3a + 7b + 16$.

Given

The initial expression: $3x^2 – 4y^2 + 5xy + 20$

The resulting expression: $- x^2 – y^2 + 6xy + 20$

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To Find

The expression that should be "taken away" (subtracted) from the initial expression to get the resulting expression.

---

Solution

Let the expression to be taken away be $P$.

According to the problem, we have:

$(3x^2 – 4y^2 + 5xy + 20) - P = (– x^2 – y^2 + 6xy + 20)$

To find $P$, we need to subtract the resulting expression from the initial expression:

$P = (3x^2 – 4y^2 + 5xy + 20) - (– x^2 – y^2 + 6xy + 20)$

Remove the parentheses, remembering to change the sign of each term in the expression being subtracted:

$P = 3x^2 – 4y^2 + 5xy + 20 - (-x^2) - (-y^2) - (6xy) - (20)$

$P = 3x^2 – 4y^2 + 5xy + 20 + x^2 + y^2 - 6xy - 20$

Now, group the like terms together:

$P = (3x^2 + x^2) + (– 4y^2 + y^2) + (5xy - 6xy) + (20 - 20)$

Combine the coefficients of the like terms:

$P = (3 + 1)x^2 + (– 4 + 1)y^2 + (5 - 6)xy + 0$

$P = 4x^2 + (– 3)y^2 + (– 1)xy$

$P = 4x^2 - 3y^2 - xy$

Therefore, the expression that should be taken away is $4x^2 - 3y^2 - xy$.

(a) From the sum of $3x – y + 11$ and $– y – 11$, subtract $3x – y – 11$.

Solution

Step 1: Find the sum of the first two expressions.

Sum = $(3x - y + 11) + (-y - 11)$

Remove parentheses:

$= 3x - y + 11 - y - 11$

Group like terms:

$= 3x + (-y - y) + (11 - 11)$

Combine coefficients:

$= 3x + (-1 - 1)y + 0$

$= 3x - 2y$

So, the sum of the first two expressions is $3x - 2y$.

Step 2: Subtract the third expression from the sum found in Step 1.

Result = (Sum from Step 1) - (Third expression)

$= (3x - 2y) - (3x - y - 11)$

Remove parentheses, changing the signs of the terms being subtracted:

$= 3x - 2y - 3x - (-y) - (-11)$

$= 3x - 2y - 3x + y + 11$

Group like terms:

$= (3x - 3x) + (-2y + y) + 11$

Combine coefficients:

$= (3 - 3)x + (-2 + 1)y + 11$

$= 0x + (-1)y + 11$

$= -y + 11$

Therefore, the final result for part (a) is $-y + 11$.

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(b) From the sum of $4 + 3x$ and $5 – 4x + 2x^2$, subtract the sum of $3x^2 – 5x$ and $–x^2 + 2x + 5$.

Solution

Step 1: Find the sum of the first pair of expressions.

Sum 1 = $(4 + 3x) + (5 - 4x + 2x^2)$

Remove parentheses and group like terms (in descending order of power):

$= 2x^2 + (3x - 4x) + (4 + 5)$

Combine coefficients:

$= 2x^2 + (3 - 4)x + 9$

$= 2x^2 - x + 9$

Step 2: Find the sum of the second pair of expressions.

Sum 2 = $(3x^2 - 5x) + (-x^2 + 2x + 5)$

Remove parentheses and group like terms:

$= (3x^2 - x^2) + (-5x + 2x) + 5$

Combine coefficients:

$= (3 - 1)x^2 + (-5 + 2)x + 5$

$= 2x^2 - 3x + 5$

Step 3: Subtract the second sum (Sum 2) from the first sum (Sum 1).

Result = (Sum 1) - (Sum 2)

$= (2x^2 - x + 9) - (2x^2 - 3x + 5)$

Remove parentheses, changing the signs of the terms in Sum 2:

$= 2x^2 - x + 9 - 2x^2 - (-3x) - 5$

$= 2x^2 - x + 9 - 2x^2 + 3x - 5$

Group like terms:

$= (2x^2 - 2x^2) + (-x + 3x) + (9 - 5)$

Combine coefficients:

$= (2 - 2)x^2 + (-1 + 3)x + 4$

$= 0x^2 + 2x + 4$

$= 2x + 4$

Therefore, the final result for part (b) is $2x + 4$.

We are asked to find the value of the given expressions when the variable $x$ is equal to $2$.

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(i) Expression: $x + 4$

Solution:

Substitute $x = 2$ into the expression:

Value $= 2 + 4$

Value $= 6$

Therefore, the value of $x + 4$ when $x=2$ is 6.

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(ii) Expression: $4x – 3$

Solution:

Substitute $x = 2$ into the expression:

Value $= 4(2) - 3$

$= 8 - 3$

$= 5$

Therefore, the value of $4x - 3$ when $x=2$ is 5.

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(iii) Expression: $19 – 5x^2$

Solution:

Substitute $x = 2$ into the expression:

Value $= 19 - 5(2)^2$

First, calculate the power:

$= 19 - 5(4)$

Next, perform the multiplication:

$= 19 - 20$

Finally, perform the subtraction:

$= -1$

Therefore, the value of $19 - 5x^2$ when $x=2$ is -1.

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(iv) Expression: $100 – 10x^3$

Solution:

Substitute $x = 2$ into the expression:

Value $= 100 - 10(2)^3$

First, calculate the power:

$= 100 - 10(8)$

Next, perform the multiplication:

$= 100 - 80$

Finally, perform the subtraction:

$= 20$

Therefore, the value of $100 - 10x^3$ when $x=2$ is 20.

We are asked to find the value of the given expressions when the variable $n$ is equal to $-2$.

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(i) Expression: $5n - 2$

Solution:

Substitute $n = -2$ into the expression:

Value $= 5(-2) - 2$

Perform the multiplication:

$= -10 - 2$

Perform the subtraction:

$= -12$

Therefore, the value of $5n - 2$ when $n=-2$ is $-12$.

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(ii) Expression: $5n^2 + 5n - 2$

Solution:

Substitute $n = -2$ into the expression:

Value $= 5(-2)^2 + 5(-2) - 2$

First, calculate the power: $(-2)^2 = 4$.

$= 5(4) + 5(-2) - 2$

Next, perform the multiplications:

$= 20 + (-10) - 2$

$= 20 - 10 - 2$

Perform the subtractions from left to right:

$= 10 - 2$

$= 8$

Therefore, the value of $5n^2 + 5n - 2$ when $n=-2$ is $8$.

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(iii) Expression: $n^3 + 5n^2 + 5n - 2$

Solution:

Substitute $n = -2$ into the expression:

Value $= (-2)^3 + 5(-2)^2 + 5(-2) - 2$

First, calculate the powers: $(-2)^3 = -8$ and $(-2)^2 = 4$.

$= (-8) + 5(4) + 5(-2) - 2$

Next, perform the multiplications:

$= -8 + 20 + (-10) - 2$

$= -8 + 20 - 10 - 2$

Perform the additions and subtractions from left to right:

$= 12 - 10 - 2$

$= 2 - 2$

$= 0$

Therefore, the value of $n^3 + 5n^2 + 5n - 2$ when $n=-2$ is $0$.

Given

The values of the variables are $a = 3$ and $b = 2$.

To Find

The value of the given expressions by substituting the values of $a$ and $b$.

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(i) Expression: $a + b$

Solution:

Substitute $a = 3$ and $b = 2$ into the expression:

Value $= 3 + 2$

Value $= 5$

Therefore, the value of $a + b$ when $a=3$ and $b=2$ is 5.

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(ii) Expression: $7a – 4b$

Solution:

Substitute $a = 3$ and $b = 2$ into the expression:

Value $= 7(3) - 4(2)$

Perform the multiplications:

$= 21 - 8$

Perform the subtraction:

$= 13$

Therefore, the value of $7a - 4b$ when $a=3$ and $b=2$ is 13.

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(iii) Expression: $a^2 + 2ab + b^2$

Solution:

Substitute $a = 3$ and $b = 2$ into the expression:

Value $= (3)^2 + 2(3)(2) + (2)^2$

Calculate the powers and the multiplication:

$= 9 + 12 + 4$

Perform the additions:

$= 21 + 4$

$= 25$

Therefore, the value of $a^2 + 2ab + b^2$ when $a=3$ and $b=2$ is 25.

Alternate Method: Recognize that $a^2 + 2ab + b^2 = (a+b)^2$.

Value $= (3+2)^2 = (5)^2 = 25$.

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(iv) Expression: $a^3 – b^3$

Solution:

Substitute $a = 3$ and $b = 2$ into the expression:

Value $= (3)^3 - (2)^3$

Calculate the powers:

$= 27 - 8$

Perform the subtraction:

$= 19$

Therefore, the value of $a^3 - b^3$ when $a=3$ and $b=2$ is 19.

Given

The value of the variable $m$ is $2$.

To Find

The value of the given expressions by substituting $m = 2$.

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(i) Expression: $m - 2$

Solution:

Substitute $m = 2$ into the expression:

Value $= 2 - 2$

Value $= 0$

Therefore, the value of $m - 2$ when $m=2$ is 0.

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(ii) Expression: $3m - 5$

Solution:

Substitute $m = 2$ into the expression:

Value $= 3(2) - 5$

Perform the multiplication:

$= 6 - 5$

Perform the subtraction:

$= 1$

Therefore, the value of $3m - 5$ when $m=2$ is 1.

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(iii) Expression: $9 - 5m$

Solution:

Substitute $m = 2$ into the expression:

Value $= 9 - 5(2)$

Perform the multiplication:

$= 9 - 10$

Perform the subtraction:

$= -1$

Therefore, the value of $9 - 5m$ when $m=2$ is -1.

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(iv) Expression: $3m^2 - 2m - 7$

Solution:

Substitute $m = 2$ into the expression:

Value $= 3(2)^2 - 2(2) - 7$

Calculate the power: $(2)^2 = 4$.

$= 3(4) - 2(2) - 7$

Perform the multiplications:

$= 12 - 4 - 7$

Perform the subtractions from left to right:

$= 8 - 7$

$= 1$

Therefore, the value of $3m^2 - 2m - 7$ when $m=2$ is 1.

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(v) Expression: $\frac{5m}{2} - 4$

Solution:

Substitute $m = 2$ into the expression:

Value $= \frac{5(2)}{2} - 4$

Perform the multiplication in the numerator:

$= \frac{10}{2} - 4$

Perform the division (or cancel the 2s):

$= 5 - 4$

Perform the subtraction:

$= 1$

Therefore, the value of $\frac{5m}{2} - 4$ when $m=2$ is 1.

Given

The value of the variable $p$ is $-2$.

To Find

The value of the given expressions by substituting $p = -2$.

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(i) Expression: $4p + 7$

Solution:

Substitute $p = -2$ into the expression:

Value $= 4(-2) + 7$

Perform the multiplication:

$= -8 + 7$

Perform the addition:

$= -1$

Therefore, the value of $4p + 7$ when $p=-2$ is -1.

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(ii) Expression: $– 3p^2 + 4p + 7$

Solution:

Substitute $p = -2$ into the expression:

Value $= -3(-2)^2 + 4(-2) + 7$

First, calculate the power: $(-2)^2 = 4$.

$= -3(4) + 4(-2) + 7$

Perform the multiplications:

$= -12 + (-8) + 7$

$= -12 - 8 + 7$

Perform the addition/subtraction from left to right:

$= -20 + 7$

$= -13$

Therefore, the value of $– 3p^2 + 4p + 7$ when $p=-2$ is -13.

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(iii) Expression: $– 2p^3 – 3p^2 + 4p + 7$

Solution:

Substitute $p = -2$ into the expression:

Value $= -2(-2)^3 - 3(-2)^2 + 4(-2) + 7$

First, calculate the powers: $(-2)^3 = -8$ and $(-2)^2 = 4$.

$= -2(-8) - 3(4) + 4(-2) + 7$

Perform the multiplications:

$= 16 - 12 + (-8) + 7$

$= 16 - 12 - 8 + 7$

Perform the addition/subtraction from left to right:

$= 4 - 8 + 7$

$= -4 + 7$

$= 3$

Therefore, the value of $– 2p^3 – 3p^2 + 4p + 7$ when $p=-2$ is 3.

Given

The value of the variable $x$ is $-1$.

To Find

The value of the given expressions by substituting $x = -1$.

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(i) Expression: $2x - 7$

Solution:

Substitute $x = -1$ into the expression:

Value $= 2(-1) - 7$

Perform the multiplication:

$= -2 - 7$

Perform the subtraction:

$= -9$

Therefore, the value of $2x - 7$ when $x=-1$ is -9.

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(ii) Expression: $– x + 2$

Solution:

Substitute $x = -1$ into the expression:

Value $= -(-1) + 2$

Simplify the double negative:

$= 1 + 2$

Perform the addition:

$= 3$

Therefore, the value of $– x + 2$ when $x=-1$ is 3.

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(iii) Expression: $x^2 + 2x + 1$

Solution:

Substitute $x = -1$ into the expression:

Value $= (-1)^2 + 2(-1) + 1$

Calculate the power: $(-1)^2 = 1$.

$= 1 + 2(-1) + 1$

Perform the multiplication:

$= 1 - 2 + 1$

Perform the addition/subtraction from left to right:

$= -1 + 1$

$= 0$

Therefore, the value of $x^2 + 2x + 1$ when $x=-1$ is 0.

(Alternatively, note that $x^2 + 2x + 1 = (x+1)^2$. Substituting $x=-1$ gives $(-1+1)^2 = 0^2 = 0$.)

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(iv) Expression: $2x^2 – x – 2$

Solution:

Substitute $x = -1$ into the expression:

Value $= 2(-1)^2 - (-1) - 2$

Calculate the power: $(-1)^2 = 1$.

$= 2(1) - (-1) - 2$

Perform the multiplication and simplify the double negative:

$= 2 + 1 - 2$

Perform the addition/subtraction from left to right:

$= 3 - 2$

$= 1$

Therefore, the value of $2x^2 – x – 2$ when $x=-1$ is 1.

Given

The values of the variables are $a = 2$ and $b = -2$.

To Find

The value of the given expressions by substituting $a = 2$ and $b = -2$.

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(i) Expression: $a^2 + b^2$

Solution:

Substitute $a = 2$ and $b = -2$ into the expression:

Value $= (2)^2 + (-2)^2$

Calculate the powers:

$= 4 + 4$

Perform the addition:

$= 8$

Therefore, the value of $a^2 + b^2$ when $a=2$ and $b=-2$ is 8.

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(ii) Expression: $a^2 + ab + b^2$

Solution:

Substitute $a = 2$ and $b = -2$ into the expression:

Value $= (2)^2 + (2)(-2) + (-2)^2$

Calculate the powers and the multiplication:

$= 4 + (-4) + 4$

$= 4 - 4 + 4$

Perform the addition/subtraction from left to right:

$= 0 + 4$

$= 4$

Therefore, the value of $a^2 + ab + b^2$ when $a=2$ and $b=-2$ is 4.

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(iii) Expression: $a^2 - b^2$

Solution:

Substitute $a = 2$ and $b = -2$ into the expression:

Value $= (2)^2 - (-2)^2$

Calculate the powers:

$= 4 - 4$

Perform the subtraction:

$= 0$

Therefore, the value of $a^2 - b^2$ when $a=2$ and $b=-2$ is 0.

Given

The values of the variables are $a = 0$ and $b = -1$.

To Find

The value of the given expressions by substituting $a = 0$ and $b = -1$.

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(i) Expression: $2a + 2b$

Solution:

Substitute $a = 0$ and $b = -1$ into the expression:

Value $= 2(0) + 2(-1)$

Perform the multiplications:

$= 0 + (-2)$

$= -2$

Therefore, the value of $2a + 2b$ when $a=0$ and $b=-1$ is -2.

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(ii) Expression: $2a^2 + b^2 + 1$

Solution:

Substitute $a = 0$ and $b = -1$ into the expression:

Value $= 2(0)^2 + (-1)^2 + 1$

Calculate the powers: $(0)^2 = 0$ and $(-1)^2 = 1$.

$= 2(0) + 1 + 1$

Perform the multiplication:

$= 0 + 1 + 1$

Perform the additions:

$= 2$

Therefore, the value of $2a^2 + b^2 + 1$ when $a=0$ and $b=-1$ is 2.

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(iii) Expression: $2a^2b + 2ab^2 + ab$

Solution:

Substitute $a = 0$ and $b = -1$ into the expression:

Value $= 2(0)^2(-1) + 2(0)(-1)^2 + (0)(-1)$

Calculate the powers: $(0)^2 = 0$ and $(-1)^2 = 1$.

$= 2(0)(-1) + 2(0)(1) + (0)(-1)$

Perform the multiplications. Since each term contains a factor of 0, each term becomes 0.

$= 0 + 0 + 0$

$= 0$

Therefore, the value of $2a^2b + 2ab^2 + ab$ when $a=0$ and $b=-1$ is 0.

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(iv) Expression: $a^2 + ab + 2$

Solution:

Substitute $a = 0$ and $b = -1$ into the expression:

Value $= (0)^2 + (0)(-1) + 2$

Calculate the power and the multiplication:

$= 0 + 0 + 2$

Perform the addition:

$= 2$

Therefore, the value of $a^2 + ab + 2$ when $a=0$ and $b=-1$ is 2.

Given

The value of the variable $x$ is $2$.

To Find

First, simplify each expression. Then, find the value of the simplified expression by substituting $x = 2$.

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(i) Expression: $x + 7 + 4(x - 5)$

Simplification:

Distribute the 4:

$= x + 7 + 4x - 4(5)$

$= x + 7 + 4x - 20$

Group like terms:

$= (x + 4x) + (7 - 20)$

Combine like terms:

$= 5x - 13$

Finding the value (substitute $x = 2$ into $5x - 13$):

Value $= 5(2) - 13$

$= 10 - 13$

$= -3$

The simplified expression is $5x - 13$ and its value when $x=2$ is -3.

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(ii) Expression: $3(x + 2) + 5x - 7$

Simplification:

Distribute the 3:

$= 3x + 3(2) + 5x - 7$

$= 3x + 6 + 5x - 7$

Group like terms:

$= (3x + 5x) + (6 - 7)$

Combine like terms:

$= 8x - 1$

Finding the value (substitute $x = 2$ into $8x - 1$):

Value $= 8(2) - 1$

$= 16 - 1$

$= 15$

The simplified expression is $8x - 1$ and its value when $x=2$ is 15.

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(iii) Expression: $6x + 5(x - 2)$

Simplification:

Distribute the 5:

$= 6x + 5x - 5(2)$

$= 6x + 5x - 10$

Combine like terms:

$= (6 + 5)x - 10$

$= 11x - 10$

Finding the value (substitute $x = 2$ into $11x - 10$):

Value $= 11(2) - 10$

$= 22 - 10$

$= 12$

The simplified expression is $11x - 10$ and its value when $x=2$ is 12.

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(iv) Expression: $4(2x - 1) + 3x + 11$

Simplification:

Distribute the 4:

$= 4(2x) - 4(1) + 3x + 11$

$= 8x - 4 + 3x + 11$

Group like terms:

$= (8x + 3x) + (-4 + 11)$

Combine like terms:

$= 11x + 7$

Finding the value (substitute $x = 2$ into $11x + 7$):

Value $= 11(2) + 7$

$= 22 + 7$

$= 29$

The simplified expression is $11x + 7$ and its value when $x=2$ is 29.

Given

The values of the variables are $x = 3$, $a = -1$, $b = -2$.

To Find

First, simplify each expression. Then, find the value of the simplified expression by substituting the given variable values.

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(i) Expression: $3x – 5 – x + 9$

Simplification:

Group like terms:

$= (3x - x) + (-5 + 9)$

Combine coefficients:

$= (3 - 1)x + 4$

$= 2x + 4$

Finding the value (substitute $x = 3$ into $2x + 4$):

Value $= 2(3) + 4$

$= 6 + 4$

$= 10$

The simplified expression is $2x + 4$ and its value is 10.

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(ii) Expression: $2 – 8x + 4x + 4$

Simplification:

Group like terms:

$= (-8x + 4x) + (2 + 4)$

Combine coefficients:

$= (-8 + 4)x + 6$

$= -4x + 6$

Finding the value (substitute $x = 3$ into $-4x + 6$):

Value $= -4(3) + 6$

$= -12 + 6$

$= -6$

The simplified expression is $-4x + 6$ and its value is -6.

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(iii) Expression: $3a + 5 – 8a + 1$

Simplification:

Group like terms:

$= (3a - 8a) + (5 + 1)$

Combine coefficients:

$= (3 - 8)a + 6$

$= -5a + 6$

Finding the value (substitute $a = -1$ into $-5a + 6$):

Value $= -5(-1) + 6$

$= 5 + 6$

$= 11$

The simplified expression is $-5a + 6$ and its value is 11.

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(iv) Expression: $10 – 3b – 4 – 5b$

Simplification:

Group like terms:

$= (-3b - 5b) + (10 - 4)$

Combine coefficients:

$= (-3 - 5)b + 6$

$= -8b + 6$

Finding the value (substitute $b = -2$ into $-8b + 6$):

Value $= -8(-2) + 6$

$= 16 + 6$

$= 22$

The simplified expression is $-8b + 6$ and its value is 22.

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(v) Expression: $2a – 2b – 4 – 5 + a$

Simplification:

Group like terms:

$= (2a + a) - 2b + (-4 - 5)$

Combine coefficients:

$= (2 + 1)a - 2b - 9$

$= 3a - 2b - 9$

Finding the value (substitute $a = -1, b = -2$ into $3a - 2b - 9$):

Value $= 3(-1) - 2(-2) - 9$

$= -3 - (-4) - 9$

$= -3 + 4 - 9$

$= 1 - 9$

$= -8$

The simplified expression is $3a - 2b - 9$ and its value is -8.

(i) If $z = 10$, find the value of $z^3 - 3(z - 10)$

Given:

The value of the variable $z$ is $10$.

The expression is $z^3 - 3(z - 10)$.

Solution:

Substitute $z = 10$ into the expression:

Value $= (10)^3 - 3(10 - 10)$

First, evaluate the expression inside the parentheses and the power:

$10 - 10 = 0$

$(10)^3 = 10 \times 10 \times 10 = 1000$

Substitute these back into the expression:

Value $= 1000 - 3(0)$

Perform the multiplication:

$= 1000 - 0$

Perform the subtraction:

$= 1000$

Therefore, the value of $z^3 - 3(z - 10)$ when $z=10$ is 1000.

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(ii) If $p = – 10$, find the value of $p^2 - 2p - 100$

Given:

The value of the variable $p$ is $-10$.

The expression is $p^2 - 2p - 100$.

Solution:

Substitute $p = -10$ into the expression:

Value $= (-10)^2 - 2(-10) - 100$

First, calculate the power: $(-10)^2 = (-10) \times (-10) = 100$.

Perform the multiplications:

$= 100 - (-20) - 100$

Simplify the double negative:

$= 100 + 20 - 100$

Perform the addition/subtraction from left to right:

$= 120 - 100$

$= 20$

Therefore, the value of $p^2 - 2p - 100$ when $p=-10$ is 20.

Given

The expression is $2x^2 + x - a$.

The value of the expression is $5$ when $x = 0$.

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To Find

The value of the constant $a$.

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Solution

We are given the equation:

$2x^2 + x - a = 5$

We need to find the value of $a$ when $x = 0$.

Substitute $x = 0$ into the equation:

$2(0)^2 + (0) - a = 5$

Calculate the terms involving $x$:

$2(0) + 0 - a = 5$

$0 + 0 - a = 5$

$-a = 5$

To find $a$, multiply both sides by $-1$:

$a = -5$

Therefore, the value of $a$ should be -5 for the expression $2x^2 + x - a$ to equal 5 when $x = 0$.

Given

The expression is $2(a^2 + ab) + 3 – ab$.

The values of the variables are $a = 5$ and $b = -3$.

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To Find

1. Simplify the given expression.

2. Find the value of the simplified expression using the given values of $a$ and $b$.

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Solution

Step 1: Simplify the expression $2(a^2 + ab) + 3 – ab$

First, distribute the 2 into the parentheses:

$= 2(a^2) + 2(ab) + 3 - ab$

$= 2a^2 + 2ab + 3 - ab$

Next, group the like terms (the terms containing $ab$):

$= 2a^2 + (2ab - ab) + 3$

Combine the like terms:

$= 2a^2 + (2 - 1)ab + 3$

$= 2a^2 + 1ab + 3$

$= 2a^2 + ab + 3$

The simplified expression is $2a^2 + ab + 3$.

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Step 2: Find the value of the simplified expression when $a = 5$ and $b = -3$

Substitute $a = 5$ and $b = -3$ into the simplified expression $2a^2 + ab + 3$:

Value $= 2(5)^2 + (5)(-3) + 3$

Calculate the power: $(5)^2 = 25$.

$= 2(25) + (5)(-3) + 3$

Perform the multiplications:

$= 50 + (-15) + 3$

$= 50 - 15 + 3$

Perform the subtraction and addition from left to right:

$= 35 + 3$

$= 38$

Therefore, the value of the expression when $a = 5$ and $b = -3$ is 38.

Given

The algebraic expressions for the number of segments required to form $n$ digits of a certain kind are:

(a) For the digit '6': The expression is $5n + 1$.

(b) For the digit '4': The expression is $3n + 1$.

(c) For the digit '8': The expression is $5n + 2$.

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To Find

The number of segments required to form $n=5$, $n=10$, and $n=100$ digits for each type (6, 4, and 8).

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Solution

(a) For the digit '6' (Expression: $5n + 1$)

Substitute the given values of $n$ into the expression $5n + 1$.

For $n = 5$ digits:

Number of segments $= 5(5) + 1 = 25 + 1 = 26$.

For $n = 10$ digits:

Number of segments $= 5(10) + 1 = 50 + 1 = 51$.

For $n = 100$ digits:

Number of segments $= 5(100) + 1 = 500 + 1 = 501$.

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(b) For the digit '4' (Expression: $3n + 1$)

Substitute the given values of $n$ into the expression $3n + 1$.

For $n = 5$ digits:

Number of segments $= 3(5) + 1 = 15 + 1 = 16$.

For $n = 10$ digits:

Number of segments $= 3(10) + 1 = 30 + 1 = 31$.

For $n = 100$ digits:

Number of segments $= 3(100) + 1 = 300 + 1 = 301$.

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(c) For the digit '8' (Expression: $5n + 2$)

Substitute the given values of $n$ into the expression $5n + 2$.

For $n = 5$ digits:

Number of segments $= 5(5) + 2 = 25 + 2 = 27$.

For $n = 10$ digits:

Number of segments $= 5(10) + 2 = 50 + 2 = 52$.

For $n = 100$ digits:

Number of segments $= 5(100) + 2 = 500 + 2 = 502$.

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Summary Table:

Digit Type	Expression	Segments for 5 digits ($n=5$)	Segments for 10 digits ($n=10$)	Segments for 100 digits ($n=100$)
6	$5n + 1$	26	51	501
4	$3n + 1$	16	31	301
8	$5n + 2$	27	52	502

To complete the table, we substitute the term number ($n = 1, 2, 3, 4, 5, 10, 100$) into the given algebraic expression for each row.

(i) Expression: $2n - 1$

For $n=5$: $2(5) - 1 = 10 - 1 = 9$.

For $n=10$: $2(10) - 1 = 20 - 1 = 19$.

For $n=100$: $2(100) - 1 = 200 - 1 = 199$.

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(ii) Expression: $3n + 2$

For $n=5$: $3(5) + 2 = 15 + 2 = 17$.

For $n=10$: $3(10) + 2 = 30 + 2 = 32$.

For $n=100$: $3(100) + 2 = 300 + 2 = 302$.

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(iii) Expression: $4n + 1$

For $n=5$: $4(5) + 1 = 20 + 1 = 21$.

For $n=10$: $4(10) + 1 = 40 + 1 = 41$.

For $n=100$: $4(100) + 1 = 400 + 1 = 401$.

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(iv) Expression: $7n + 20$

For $n=5$: $7(5) + 20 = 35 + 20 = 55$.

For $n=10$: $7(10) + 20 = 70 + 20 = 90$.

For $n=100$: $7(100) + 20 = 700 + 20 = 720$.

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(v) Expression: $n^2 + 1$

For $n=5$: $(5)^2 + 1 = 25 + 1 = 26$.

For $n=10$: $(10)^2 + 1 = 100 + 1 = 101$.

For $n=100$: $(100)^2 + 1 = 10000 + 1 = 10001$.

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The completed table is as follows:

S. No.	Expression	Terms
		1st	2nd	3rd	4th	5th	...	10th	...	100th	...
(i)	$2n – 1$	1	3	5	7	9	...	19	...	199	...
(ii)	$3n + 2$	5	8	11	14	17	...	32	...	302	...
(iii)	$4n + 1$	5	9	13	17	21	...	41	...	401	...
(iv)	$7n + 20$	27	34	41	48	55	...	90	...	720	...
(v)	$n^2 + 1$	2	5	10	17	26	...	101	...	10,001	...