Linear Equations in Two Variables Class 9: Graphs & Solutions

A linear equation in two variables is an equation that can be written in the standard form:

The word 'linear' comes from 'line' — the graph of every such equation is a straight line on the Cartesian plane.

Examples:
- $2x + 3y = 12$ can be written as $2x + 3y - 12 = 0$ (here $a = 2$, $b = 3$, $c = -12$)
- $x - y = 0$ (here $a = 1$, $b = -1$, $c = 0$)
- $y = 5$ can be written as $0 \cdot x + 1 \cdot y - 5 = 0$

Not a linear equation: $xy = 6$ (product of variables), $x^2 + y = 3$ (variable squared).

A solution of $ax + by + c = 0$ is a pair of values $(x, y)$ that makes the equation true.

Key Fact: A linear equation in two variables has infinitely many solutions. Each solution is an ordered pair $(x, y)$ that lies on the line.

How to find solutions: Pick a value for one variable, substitute it into the equation and solve for the other.

Solved Example:
Find three solutions of $2x + 3y = 12$.

Solution:
Rewrite as $y = \frac{12 - 2x}{3}$.

All three pairs satisfy the equation. There are infinitely many more — every point on the line $2x + 3y = 12$ is a solution.

The graph of $ax + by + c = 0$ is always a straight line. Here is the step-by-step process to draw it.

Step 1: Find at least two solutions (three is better for accuracy).
Step 2: Plot the points on the Cartesian plane.
Step 3: Join the points with a straight line. Extend the line in both directions with arrows.

Solved Example:
Draw the graph of $x + 2y = 6$.

Rewrite as $y = \frac{6 - x}{2}$.

Plot $(0, 3)$, $(2, 2)$, $(4, 1)$ and $(6, 0)$ on graph paper. Join them — you get a straight line.

Important Observation: Every point on this line is a solution of $x + 2y = 6$, and every solution of the equation lies on this line. The line is the geometric representation of the equation.

These are special cases that students often find tricky but are actually very simple.

1. Line parallel to the x-axis:
The equation is of the form $y = k$ (where $k$ is a constant).
- Every point on this line has the same $y$-coordinate.
- Example: $y = 3$ is a horizontal line passing through $(0, 3)$, $(1, 3)$, $(-2, 3)$, etc.
- The x-axis itself has the equation $y = 0$.

2. Line parallel to the y-axis:
The equation is of the form $x = k$ (where $k$ is a constant).
- Every point on this line has the same $x$-coordinate.
- Example: $x = -2$ is a vertical line passing through $(-2, 0)$, $(-2, 1)$, $(-2, -3)$, etc.
- The y-axis itself has the equation $x = 0$.

Solved Example:
Write the equation of the line passing through $(3, 5)$ and parallel to the y-axis.

Solution: A line parallel to the y-axis has the equation $x = k$. Since it passes through $(3, 5)$, we have $k = 3$.

The equation is $x = 3$.

Many NCERT questions ask you to form an equation from a real-life situation and then graph it.

Example 1:
The cost of a notebook is twice the cost of a pen. Write a linear equation in two variables to represent this and draw its graph.

Solution:
Let cost of a pen $= x$ (Rs) and cost of a notebook $= y$ (Rs).

Given: $y = 2x$, or equivalently $2x - y = 0$.

Plot $(1, 2)$, $(2, 4)$, $(3, 6)$ and join them. The line passes through the origin.

Example 2:
The perimeter of a rectangle is $40 \text{ cm}$. Express this as a linear equation and find three possible dimensions.

Solution:
Let length $= x$ and breadth $= y$.

Since $x > 0$ and $y > 0$, the graph is the portion of the line $x + y = 20$ in the first quadrant.

Here are important facts that examiners love to test.

1. Every linear equation in two variables has infinitely many solutions. Its graph is a straight line extending infinitely in both directions.

**2. A linear equation in one variable ($ax + b = 0$) can be viewed in two ways:**
- On a number line: a single point $x = -b/a$.
- On the Cartesian plane: a vertical line $x = -b/a$ (parallel to y-axis).

**3. The equation $y = mx$ always passes through the origin** $(0, 0)$. The value $m$ determines the slope (steepness) of the line.

4. Intercepts:
- x-intercept: The point where the line crosses the x-axis. Set $y = 0$ and solve for $x$.
- y-intercept: The point where the line crosses the y-axis. Set $x = 0$ and solve for $y$.

Solved Example:
Find the x-intercept and y-intercept of $3x + 4y = 24$.

For x-intercept: $y = 0 \Rightarrow 3x = 24 \Rightarrow x = 8$. Point: $(8, 0)$.
For y-intercept: $x = 0 \Rightarrow 4y = 24 \Rightarrow y = 6$. Point: $(0, 6)$.

Example 1: Check whether $(2, 3)$ is a solution of $5x - 2y = 4$.

Solution:
Substitute $x = 2$, $y = 3$:

Example 2: The equation $2x + 3y = k$ passes through the point $(4, -1)$. Find $k$.

Solution:
Substitute $x = 4$, $y = -1$:

Key Takeaways:

1. The standard form is $ax + by + c = 0$ with $a$ and $b$ not both zero.
2. Every linear equation in two variables has infinitely many solutions — each is a point on the line.
3. To draw the graph: find 2-3 solutions, plot them, join with a straight line.
4. $y = k$ is a horizontal line; $x = k$ is a vertical line.
5. The x-intercept is found by setting $y = 0$; the y-intercept by setting $x = 0$.

Connection to Class 10: In Class 10 (Chapter 3), you will study pairs of linear equations — two lines on the same plane — and learn about consistent, inconsistent and dependent systems. Mastering single linear equations now makes that chapter much easier.

Ready to practise? Head to the SparkEd Linear Equations practice page for adaptive problems with instant feedback. Use the SparkEd Math Solver to check your graphs or ask the SparkEd Coach for a walkthrough of any problem.