| Classwise Concept with Examples | ||||||
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| 6th | 7th | 8th | 9th | 10th | 11th | 12th |
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| Linear Equations in Two Variables - Introduction and Formation | Graph of a Linear Equation in Two Variables | Equations of a Lines Parallel to x-axis and y-axis |
Chapter 4 Linear Equations In Two Variables (Concepts)
Having mastered the techniques for solving linear equations involving a single variable in Class 8, we now embark on a significant expansion of our algebraic toolkit by introducing Linear Equations in Two Variables. This chapter shifts our focus from finding a single numerical solution to understanding relationships between two quantities, typically represented by the variables $x$ and $y$. These equations form the foundation for modeling a vast array of real-world situations where two quantities change in relation to each other.
We formally define a linear equation in two variables as any equation that can be expressed in the standard form: $$ ax + by + c = 0 $$ where $a$, $b$, and $c$ are real numbers, and critically, $a$ and $b$ are not both zero simultaneously (if both were zero, the variables would vanish). The term 'linear' signifies that both variables $x$ and $y$ appear only with an exponent of 1 (no $x^2$, $y^3$, $xy$, etc.). Examples include equations like $2x + 3y = 5$ (which can be written as $2x + 3y - 5 = 0$) or $y = 4x - 1$ (rewritten as $4x - y - 1 = 0$). A fundamental departure from single-variable equations is that these two-variable linear equations possess infinitely many solutions. A solution is no longer a single number but an ordered pair of values, $(x, y)$, which, when substituted into the equation, makes the statement true. For example, in the equation $x + y = 7$, the pair $(3, 4)$ is a solution because $3 + 4 = 7$. But so are $(0, 7)$, $(7, 0)$, $(-1, 8)$, $(3.5, 3.5)$, and countless others. We learn to find these solutions by arbitrarily choosing a value for one variable (say $x$) and then solving the resulting simple linear equation for the other variable ($y$), or vice versa.
A major focus of this chapter lies in the graphical representation of these equations. Leveraging the Cartesian coordinate system learned previously, we discover a profound connection: the graph of every linear equation in two variables is a straight line. This visual representation provides immense insight into the nature of the equation's infinite solutions. To draw the graph of a given linear equation:
- Find at least two distinct solutions (ordered pairs $(x, y)$) to the equation.
- Plot these points accurately on the Cartesian plane.
- Draw the unique straight line that passes through these plotted points. (Finding a third solution and plotting it serves as an excellent check – if it doesn't lie on the same line, a calculation error likely occurred).
Conversely, every single point lying on this infinite straight line corresponds to an ordered pair $(x, y)$ that is a solution to the equation, and every solution pair $(x, y)$ corresponds to a point situated precisely on that line. This establishes a perfect one-to-one correspondence between the algebraic solutions and the geometric points on the line.
We also examine special cases of linear equations whose graphs are parallel to the coordinate axes:
- An equation of the form $x = k$ (where $k$ is a constant) represents a vertical line parallel to the y-axis, passing through the point $(k, 0)$.
- An equation of the form $y = k$ (where $k$ is a constant) represents a horizontal line parallel to the x-axis, passing through the point $(0, k)$.
Linear Equations in Two Variables - Introduction and Formation
In your previous classes, you learned about linear equations in one variable, such as $2x+5=0$ or $y-7=12$. These equations involve only one variable, and the highest power of that variable is 1. You also learned how to solve these equations to find a unique value for the variable that satisfies the equation. For example, the equation $2x+5=0$ has the unique solution $x = -\frac{5}{2}$.
Now, we will extend the concept of linear equations to include equations that involve two different variables. These equations are fundamental in algebra and are used to represent relationships between two varying quantities.
Introduction to Linear Equations in Two Variables
An equation is a mathematical statement that asserts the equality of two expressions. A linear equation is an equation where the highest power of each variable involved is 1, and there are no products of variables (like $xy$ or $x^2$).
A Linear Equation in Two Variables is an equation that contains exactly two variables, and the highest power of each variable is 1. It can be written in a general or standard form.
The standard form of a linear equation in two variables, typically $x$ and $y$, is:
$ax + by + c = 0$
... (i)
where:
- $x$ and $y$ are the two variables.
- $a$, $b$, and $c$ are real numbers (these can be integers, fractions, decimals, or irrational numbers like $\sqrt{2}$). $a, b,$ and $c$ are the coefficients and the constant term.
- The coefficients $a$ and $b$ are not both zero ($a \neq 0$ or $b \neq 0$). This condition ensures that both variables are present in the equation (at least one of $ax$ or $by$ must be present for it to be an equation in potentially two variables) and that the highest power of any variable is 1. If $a=0$ but $b \neq 0$, the equation becomes $by+c=0$, which is a linear equation in one variable $y$. Similarly, if $b=0$ but $a \neq 0$, the equation is $ax+c=0$, a linear equation in one variable $x$. If both $a$ and $b$ are non-zero, it's explicitly a linear equation in two variables.
The variables can be any two distinct letters, such as $p$ and $q$, $u$ and $v$, etc. The standard form would then be $ap + bq + c = 0$ or $au + bv + c = 0$, depending on the variables used.
Examples of Linear Equations in Two Variables:
- $2x + 3y = 5$: This is a linear equation in variables $x$ and $y$. The highest power of $x$ is 1, and the highest power of $y$ is 1. Coefficients of $x$ and $y$ (2 and 3) are non-zero.
- $x - 2y - 4 = 0$: A linear equation in $x$ and $y$. Highest powers are 1. Coefficients are 1 and -2 (both non-zero).
- $y = 2x - 1$: This can be rewritten by moving all terms to one side: $2x - y - 1 = 0$. This is in the standard form $ax+by+c=0$ with $a=2, b=-1, c=-1$.
- $5x = 7$: This equation only explicitly shows one variable, $x$. However, it can be written in the standard form with $y$ by including a $0y$ term: $5x + 0y - 7 = 0$. This is a linear equation in two variables where the coefficient of $y$ is zero.
- $-3y = 8$: This can be written as $0x - 3y - 8 = 0$. It is a linear equation in two variables where the coefficient of $x$ is zero.
- $\sqrt{2}x + \frac{1}{3}y = \sqrt{3}$: This can be written as $\sqrt{2}x + \frac{1}{3}y - \sqrt{3} = 0$. The coefficients $\sqrt{2}$, $\frac{1}{3}$, and the constant term $-\sqrt{3}$ are real numbers. The highest powers of $x$ and $y$ are 1.
Forming and Writing Linear Equations in Two Variables
Given a linear equation, we can arrange its terms to write it in the standard form $ax + by + c = 0$ and identify the values of $a$, $b$, and $c$. This involves moving all terms to one side of the equality sign such that the other side is 0.
Example 1. Write the equation $2x + 3y = 5$ in the form $ax+by+c=0$ and find the values of $a, b, c$.
Answer:
Given equation: $2x + 3y = 5$.
To write this in the standard form $ax+by+c=0$, we need to bring all terms to the left side such that the right side is 0. Subtract 5 from both sides of the equation:
$2x + 3y - 5 = 5 - 5$
$2x + 3y - 5 = 0$
This equation is now in the standard form $ax + by + c = 0$.
Comparing $2x + 3y - 5 = 0$ with $ax + by + c = 0$, we can identify the coefficients and the constant term:
- The coefficient of $x$ is $a = 2$.
- The coefficient of $y$ is $b = 3$.
- The constant term is $c = -5$.
So, the standard form of $2x+3y=5$ is $2x+3y-5=0$ with $a=2$, $b=3$, and $c=-5$.
Example 2. Write the equation $x - 4 = \sqrt{3}y$ in the form $ax+by+c=0$ and find the values of $a, b, c$.
Answer:
Given equation: $x - 4 = \sqrt{3}y$.
To write this in the standard form $ax+by+c=0$, move all terms to one side. Subtract $\sqrt{3}y$ from both sides:
$x - 4 - \sqrt{3}y = \sqrt{3}y - \sqrt{3}y$
$x - \sqrt{3}y - 4 = 0$
This equation is now in the standard form $ax + by + c = 0$.
Comparing $x - \sqrt{3}y - 4 = 0$ with $ax + by + c = 0$, we have:
- The coefficient of $x$ is $a = 1$.
- The coefficient of $y$ is $b = -\sqrt{3}$.
- The constant term is $c = -4$.
So, the standard form of $x - 4 = \sqrt{3}y$ is $x - \sqrt{3}y - 4 = 0$ with $a=1$, $b=-\sqrt{3}$, and $c=-4$. Note that the coefficients can be irrational numbers.
Example 3. Write the equation $5x = -3y$ in the form $ax+by+c=0$ and find the values of $a, b, c$.
Answer:
Given equation: $5x = -3y$.
To write this in the standard form $ax+by+c=0$, move all terms to one side. Add $3y$ to both sides:
$5x + 3y = -3y + 3y$
$5x + 3y = 0$
This equation can be explicitly written in the standard form $ax+by+c=0$ by including a zero constant term:
$5x + 3y + 0 = 0$
Comparing $5x + 3y + 0 = 0$ with $ax + by + c = 0$, we have:
- The coefficient of $x$ is $a = 5$.
- The coefficient of $y$ is $b = 3$.
- The constant term is $c = 0$.
So, the standard form of $5x = -3y$ is $5x+3y+0=0$ with $a=5$, $b=3$, and $c=0$. Note that the constant term can be zero.
Solutions of a Linear Equation in Two Variables
A solution of a linear equation in two variables $ax+by+c=0$ is a pair of values, one for $x$ and one for $y$, that makes the equation true. When you substitute these values into the equation, the Left Hand Side (LHS) equals the Right Hand Side (RHS).
A solution is usually written as an ordered pair $(x, y)$, where the value for $x$ is listed first, followed by the value for $y$.
Unlike linear equations in one variable, which typically have a single solution, a linear equation in two variables has infinitely many solutions.
To find solutions, you can choose any real number value for one of the variables and then solve the equation for the other variable.
Example: Consider the equation $x+y=5$. Let's find some solutions:
- If $x=1$, substitute into the equation: $1+y=5 \implies y=5-1 \implies y=4$. The ordered pair $(1, 4)$ is a solution. Check: $1+4=5$, True.
- If $x=2$, substitute: $2+y=5 \implies y=5-2 \implies y=3$. The ordered pair $(2, 3)$ is a solution. Check: $2+3=5$, True.
- If $x=0$, substitute: $0+y=5 \implies y=5$. The ordered pair $(0, 5)$ is a solution. Check: $0+5=5$, True.
- If $y=0$, substitute: $x+0=5 \implies x=5$. The ordered pair $(5, 0)$ is a solution. Check: $5+0=5$, True.
- If $x=-1$, substitute: $-1+y=5 \implies y=5+1 \implies y=6$. The ordered pair $(-1, 6)$ is a solution. Check: $-1+6=5$, True.
- If $x=10$, substitute: $10+y=5 \implies y=5-10 \implies y=-5$. The ordered pair $(10, -5)$ is a solution. Check: $10+(-5)=5$, True.
- If $x=\sqrt{2}$, substitute: $\sqrt{2}+y=5 \implies y=5-\sqrt{2}$. The ordered pair $(\sqrt{2}, 5-\sqrt{2})$ is a solution.
Since there are infinitely many real numbers we can choose for $x$ (or $y$), there are infinitely many corresponding values for $y$ (or $x$). Each such ordered pair $(x, y)$ is a solution. Therefore, a linear equation in two variables has infinitely many solutions.
Geometrically, the set of all solutions to a linear equation in two variables forms a straight line in the Cartesian plane, which is why it is called a 'linear' equation.
Graph of a Linear Equation in Two Variables
We know that a linear equation in two variables has infinitely many solutions. Each solution is an ordered pair of numbers $(x, y)$ that satisfies the equation. Since each ordered pair $(x, y)$ corresponds to a unique point in the Cartesian plane, we can represent the solutions of a linear equation as points on the plane. When we plot these points, we observe a distinct pattern.
The Graph of a Linear Equation is a Straight Line
If you take several solutions of a linear equation in two variables and plot them as points on the Cartesian plane, you will find that all these points lie on the same straight line. This is the fundamental geometric representation of a linear equation in two variables.
The set of all points $(x, y)$ in the plane that satisfy the equation $ax+by+c=0$ forms a straight line. Conversely, every point located on the straight line representing the graph of $ax+by+c=0$ is a solution to the equation.
This property is precisely why equations of the form $ax+by+c=0$ (where $a$ and $b$ are not both zero) are called 'linear' equations – their graph is a line.
Steps to Draw the Graph of a Linear Equation
To draw the graph of a linear equation in two variables ($ax+by+c=0$):
- Find Solutions: Find at least two, but preferably three, distinct solutions $(x, y)$ for the equation. To find a solution, choose a convenient real number value for one variable (say $x$) and substitute it into the equation to solve for the corresponding value of the other variable (say $y$). Repeat this process to get multiple ordered pairs. Choosing $x=0$ and solving for $y$, and choosing $y=0$ and solving for $x$ often gives the points where the line intersects the axes, which are easy to plot.
- Create a Table of Solutions: Organise the solutions you found in a table with columns for $x$ and $y$. This helps in keeping track of the ordered pairs $(x, y)$.
- Plot the Points: Draw the Cartesian coordinate axes (x-axis and y-axis) on a graph sheet. Choose appropriate scales for both axes based on the range of values in your solutions. Plot the ordered pairs from your table as points on the Cartesian plane.
- Draw the Line: Use a ruler to draw a straight line that passes through all the plotted points. Extend the line beyond the plotted points in both directions and draw arrows on both ends to show that the line continues infinitely.
- Label the Graph: Write the equation on the line itself or near the line to identify which equation the graph represents.
Using three points to draw the line is recommended. If the three points are collinear, it confirms that your solutions were calculated correctly. If they do not lie on a single straight line, there is an error in finding at least one of the solutions, and you should recheck your calculations.
Example 1. Draw the graph of the equation $x+y=4$.
Answer:
Given equation: $x+y=4$. We need to find at least two solutions. Let's find three.
Step 1: Find solutions. It's easy to find values for $y$ by choosing values for $x$ (or vice versa).
- Let $x=0$: Substitute into the equation: $0+y=4 \implies y=4$. Solution: $(0, 4)$.
- Let $x=4$: Substitute into the equation: $4+y=4 \implies y=0$. Solution: $(4, 0)$.
- Let $x=2$: Substitute into the equation: $2+y=4 \implies y=2$. Solution: $(2, 2)$.
Step 2: Create a table of solutions:
| x | y |
|---|---|
| 0 | 4 |
| 4 | 0 |
| 2 | 2 |
Step 3: Plot the points (0, 4), (4, 0), and (2, 2) on the Cartesian plane. Draw x and y axes and choose a scale where these points can be conveniently plotted (e.g., 1 unit per box).
Step 4: Draw a straight line passing through these three points. Extend the line and add arrows.
Step 5: Label the line as $x+y=4$.
This straight line is the graph of the equation $x+y=4$. Every point on this line is a solution to the equation, and every solution to the equation lies on this line.
Example 2. Draw the graph of the equation $y = 2x - 1$.
Answer:
Given equation: $y = 2x - 1$. It is already in a form that makes it easy to find $y$ for chosen values of $x$. Let's find three solutions.
Step 1: Find solutions.
- Let $x=0$: $y = 2(0) - 1 = 0 - 1 = -1$. Solution: $(0, -1)$.
- Let $x=1$: $y = 2(1) - 1 = 2 - 1 = 1$. Solution: $(1, 1)$.
- Let $x=-1$: $y = 2(-1) - 1 = -2 - 1 = -3$. Solution: $(-1, -3)$.
Step 2: Table of solutions:
| x | y |
|---|---|
| 0 | -1 |
| 1 | 1 |
| -1 | -3 |
Step 3: Plot the points (0, -1), (1, 1), and (-1, -3) on the Cartesian plane. Choose scales to accommodate these points (e.g., 1 unit per box).
Step 4: Draw a straight line passing through these three points. Extend and add arrows.
Step 5: Label the line as $y = 2x - 1$.
This straight line is the graph of the equation $y=2x-1$.
Example 3. Draw the graph of the equation $2x + y = 6$.
Answer:
Given equation: $2x + y = 6$. Let's find three solutions. It might be easier to solve for $y$: $y = 6 - 2x$.
Step 1: Find solutions.
- Let $x=0$: $y = 6 - 2(0) = 6 - 0 = 6$. Solution: $(0, 6)$.
- Let $x=3$: $y = 6 - 2(3) = 6 - 6 = 0$. Solution: $(3, 0)$.
- Let $x=1$: $y = 6 - 2(1) = 6 - 2 = 4$. Solution: $(1, 4)$.
Step 2: Table of solutions:
| x | y |
|---|---|
| 0 | 6 |
| 3 | 0 |
| 1 | 4 |
Step 3: Plot the points (0, 6), (3, 0), and (1, 4) on the Cartesian plane. Choose scales to fit these points (e.g., 1 unit per box, but y-axis may need to go up to 6).
Step 4: Draw a straight line passing through these three points. Extend and add arrows.
Step 5: Label the line as $2x+y=6$.
This straight line is the graph of the equation $2x+y=6$.
Equations of Lines Parallel to x-axis and y-axis
We have learned that the graph of any linear equation in two variables $ax+by+c=0$ is a straight line. Some special cases of linear equations result in lines that are parallel to the coordinate axes (the x-axis or the y-axis). These equations have particularly simple forms.
Equation of a Line Parallel to the x-axis
Consider a straight line that is parallel to the x-axis. What property do all the points on such a line share? If a line is parallel to the x-axis, the vertical distance of every point on that line from the x-axis is the same. This vertical distance is given by the y-coordinate.
So, for any point $(x, y)$ on a line parallel to the x-axis, the y-coordinate ($y$) is always a constant value. The x-coordinate ($x$) can vary freely along the line.
If the line is at a distance of $k$ units from the x-axis:
- If the line is above the x-axis, the y-coordinate of every point on the line is $k$ (where $k > 0$). The equation of the line is $y = k$.
- If the line is below the x-axis, the y-coordinate of every point on the line is $-k$ (where $k > 0$). The equation of the line is $y = -k$.
- If the line is the x-axis itself, the y-coordinate of every point is 0. The equation is $y = 0$.
In general, the equation of a line parallel to the x-axis is of the form $y = k$, where $k$ is a constant (any real number). This can be written in the standard linear equation form $ax+by+c=0$ as $0 \cdot x + 1 \cdot y - k = 0$, where $a=0, b=1, c=-k$. Since $b=1 \neq 0$, it is a linear equation in two variables.
Example 1. Draw the graph of the equation $y=3$.
Answer:
Given equation: $y=3$. This is a linear equation in two variables, $0x + 1y = 3$ or $0x + y - 3 = 0$.
The equation tells us that for any value of $x$, the value of $y$ is always 3. Let's find a few solutions by choosing some values for $x$:
- If $x=0$, $y=3$. Solution: $(0, 3)$.
- If $x=1$, $y=3$. Solution: $(1, 3)$.
- If $x=-2$, $y=3$. Solution: $(-2, 3)$.
- If $x=5$, $y=3$. Solution: $(5, 3)$.
Table of solutions:
| x | y |
|---|---|
| 0 | 3 |
| 1 | 3 |
| -2 | 3 |
| 5 | 3 |
Plot these points on the Cartesian plane. Draw x and y axes and choose scales. Plot (0, 3), (1, 3), (-2, 3), (5, 3).
All these points lie on a horizontal line where the y-coordinate is always 3.
Draw a straight line passing through these points. This line is parallel to the x-axis and is located 3 units above the x-axis.
This is the graph of $y=3$.
The equation of the x-axis itself is $y=0$, as every point on the x-axis has a y-coordinate of 0 (e.g., (1, 0), (-5, 0), (10, 0)).
Equation of a Line Parallel to the y-axis
Consider a straight line that is parallel to the y-axis. What property do all the points on such a line share? If a line is parallel to the y-axis, the horizontal distance of every point on that line from the y-axis is the same. This horizontal distance is given by the x-coordinate.
So, for any point $(x, y)$ on a line parallel to the y-axis, the x-coordinate ($x$) is always a constant value. The y-coordinate ($y$) can vary freely along the line.
If the line is at a distance of $k$ units from the y-axis:
- If the line is to the right of the y-axis, the x-coordinate of every point on the line is $k$ (where $k > 0$). The equation of the line is $x = k$.
- If the line is to the left of the y-axis, the x-coordinate of every point on the line is $-k$ (where $k > 0$). The equation of the line is $x = -k$.
- If the line is the y-axis itself, the x-coordinate of every point is 0. The equation is $x = 0$.
In general, the equation of a line parallel to the y-axis is of the form $x = k$, where $k$ is a constant (any real number). This can be written in the standard linear equation form $ax+by+c=0$ as $1 \cdot x + 0 \cdot y - k = 0$, where $a=1, b=0, c=-k$. Since $a=1 \neq 0$, it is a linear equation in two variables.
Example 2. Draw the graph of the equation $x=-2$.
Answer:
Given equation: $x=-2$. This is a linear equation in two variables, $1x + 0y = -2$ or $x + 2 = 0$ or $x + 0y + 2 = 0$.
The equation tells us that for any value of $y$, the value of $x$ is always -2. Let's find a few solutions by choosing some values for $y$:
- If $y=0$, $x=-2$. Solution: $(-2, 0)$.
- If $y=1$, $x=-2$. Solution: $(-2, 1)$.
- If $y=-3$, $x=-2$. Solution: $(-2, -3)$.
- If $y=10$, $x=-2$. Solution: $(-2, 10)$.
Table of solutions:
| x | y |
|---|---|
| -2 | 0 |
| -2 | 1 |
| -2 | -3 |
| -2 | 10 |
Plot these points on the Cartesian plane. Draw x and y axes and choose scales. Plot (-2, 0), (-2, 1), (-2, -3), (-2, 10).
All these points lie on a vertical line where the x-coordinate is always -2.
Draw a straight line passing through these points. This line is parallel to the y-axis and is located 2 units to the left of the y-axis.
This is the graph of $x=-2$.
The equation of the y-axis itself is $x=0$, as every point on the y-axis has an x-coordinate of 0 (e.g., (0, 1), (0, -5), (0, 100)).
Understanding the equations of lines parallel to the axes is a useful step before exploring the graphs of more general linear equations.