CBSE Class 9 Maths — Linear Equations in Two Variables

Chapter Overview & Weightage

Linear Equations in Two Variables is Chapter 4 in CBSE Class 9 Maths. This chapter bridges the gap between Class 8 algebra (single-variable equations) and the coordinate geometry that comes in Chapter 3. Understanding this chapter well makes graphing in subsequent chapters feel natural.

Exam Year	Marks Allocated	Question Types
2024	8–10 marks	1 MCQ + 1 short + 1 long
2023	6–8 marks	1 short + 1 long
2022	6–8 marks	1 short + 1 graph question
2021	8 marks	2 short + 1 long
2020	6 marks	1 short + 1 long

This chapter consistently contributes 6–10 marks in the CBSE Class 9 Annual Exam. The graph-plotting question (where you draw the line from two solutions) appears almost every year. Mastering the graphical representation earns easy full marks.

Key Concepts You Must Know

Linear equation in two variables: An equation of the form $ax + by + c = 0$ where $a, b, c$ are real numbers and $a, b$ are not both zero. Example: $2x + 3y - 6 = 0$ .
Solution of a linear equation: An ordered pair $(x, y)$ that satisfies the equation. Every linear equation in two variables has infinitely many solutions — they form a straight line.
Graph of a linear equation: Always a straight line. To draw it, find at least two solutions (ordered pairs), plot them, and join with a line extended in both directions.
x-axis equation: $y = 0$ represents the x-axis. Any equation of the form $y = k$ (horizontal line).
y-axis equation: $x = 0$ represents the y-axis. Any equation of the form $x = k$ (vertical line).
Equations of lines parallel to axes: $y = 3$ is a horizontal line 3 units above x-axis; $x = -2$ is a vertical line 2 units left of y-axis.
Linear equation as function of one variable: A linear equation $ax + b = 0$ (no $y$ term) can be written as $x = -b/a$ — a vertical line in the coordinate plane.

Important Formulas

ax + by + c = 0

where $a, b, c$ are real numbers; $a$ and $b$ are not both zero.

To find solutions: Fix any value of $x$ , solve for $y$ (or vice versa).

Example: $2x + 3y = 12$

Set $x = 0$ : $y = 4$ → Solution $(0, 4)$

Set $y = 0$ : $x = 6$ → Solution $(6, 0)$

These two points (x-intercept and y-intercept) are enough to draw the line.

$y = mx$ : Line through origin with slope $m$

$y = c$ : Horizontal line at height $c$ (parallel to x-axis)

$x = d$ : Vertical line at distance $d$ (parallel to y-axis)

$y = 0$ : The x-axis itself

$x = 0$ : The y-axis itself

Solved Previous Year Questions

PYQ 1: (CBSE 2023, 3 marks)

Q: Express $y$ in terms of $x$ for the equation $3x - 5y = 15$ . Check whether the points $(5, 0)$ and $(0, -3)$ are solutions.

Solution:

Rearranging: $5y = 3x - 15$ , so $y = \frac{3x - 15}{5} = \frac{3(x-5)}{5}$

Check $(5, 0)$ : LHS = $3(5) - 5(0) = 15$ = RHS ✓ — it is a solution.

Check $(0, -3)$ : LHS = $3(0) - 5(-3) = 15$ = RHS ✓ — it is a solution.

Both points satisfy the equation, confirming they are on the line $3x - 5y = 15$ .

PYQ 2: (CBSE 2024, 4 marks)

Q: Draw the graph of $2x + y = 6$ . Shade the region bounded by the line and the coordinate axes. Find the area of the shaded region.

Solution:

Finding intercepts:

Set $x = 0$ : $y = 6$ → point $(0, 6)$
Set $y = 0$ : $x = 3$ → point $(3, 0)$

Draw the line through $(0, 6)$ and $(3, 0)$ .

The shaded region (triangle formed by the line and both axes) has:

Base = 3 units (along x-axis, from origin to $(3,0)$ )
Height = 6 units (along y-axis, from origin to $(0,6)$ )

\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 3 \times 6 = 9 \text{ sq. units}

PYQ 3: (CBSE 2022, 2 marks)

Q: Write the equation of a line passing through $(-3, 4)$ and having slope $\frac{2}{3}$ .

Solution:

The slope-intercept form is $y = mx + c$ . With slope $m = 2/3$ :

$y = \frac{2}{3}x + c$

Substituting $(-3, 4)$ : $4 = \frac{2}{3}(-3) + c = -2 + c$ , so $c = 6$ .

Equation: $y = \frac{2}{3}x + 6$ , or multiplying through by 3: $\mathbf{2x - 3y + 18 = 0}$

Difficulty Distribution

Difficulty	Type	What to Expect
Easy (40%)	Verify solutions, write equations in standard form	Direct substitution and rearrangement
Medium (40%)	Draw graph of equation, find intercepts	Plotting 2-3 points, connecting the line
Hard (20%)	Find area of region bounded by line and axes	Combine graphing + area of triangle formula

Expert Strategy

Always find x-intercept and y-intercept first. For any linear equation, set $y = 0$ to get the x-intercept and $x = 0$ to get the y-intercept. These two points are almost always sufficient to draw the graph accurately. Finding a third point as a check is good practice for board exams.

For graph questions: Use a ruler, mark points precisely with a sharp pencil, and extend the line slightly beyond the plotted points in both directions. Examiners deduct marks for lines that don’t extend properly. Label the line with its equation.

When a problem says “write 3 solutions,” don’t just write any random pairs — choose values of $x$ that give whole number values of $y$ . For $2x + 3y = 12$ : $x = 0 \to y = 4$ , $x = 3 \to y = 2$ , $x = 6 \to y = 0$ . Clean numbers make graphs easier and reduce arithmetic errors.

For verification problems, always substitute the ordered pair into the original equation, compute both sides, and explicitly state “LHS = RHS, hence (a, b) is a solution.”

Common Traps

Trap 1: Confusing “the equation has no solution” with “the equation has one solution.” A linear equation in two variables always has infinitely many solutions — you can always find a pair $(x, y)$ that satisfies it. The concept of “no solution” or “unique solution” arises only when you have a system of two equations (which is Class 10 content).

Trap 2: Equations of lines parallel to axes. Students often confuse which type of equation gives a horizontal vs vertical line. Remember: $y = k$ (constant) → horizontal line (y doesn’t change as x changes). $x = k$ (constant) → vertical line (x doesn’t change as y changes). The equation $x = 3$ is a vertical line passing through all points where the x-coordinate is 3.

Trap 3: Forgetting that $ax + b = 0$ in one variable is a special case. The equation $2x + 4 = 0$ (or $x = -2$ ) represents a vertical line in the xy-plane, not just a single point. In one-variable thinking, $x = -2$ is a point on a number line. In two-variable coordinate geometry, it’s the vertical line $x = -2$ .