CBSE Class 10 Maths — Pair of Linear Equations

Chapter Overview & Weightage

Pair of Linear Equations is a scoring chapter that combines algebra with geometry. The graphical interpretation — two lines intersecting, parallel, or coincident — makes the abstract concrete.

Weightage: This chapter typically carries 8-10 marks in CBSE Class 10 boards. Expect a 4-mark long-answer algebraic solution (substitution or elimination), a 2-mark graphical question, and possibly a 2-mark word problem.

Year	Marks	Topics
2024	8	Elimination + word problem
2023	10	Cross-multiplication, graphical method
2022	8	Substitution, age problem

Key Concepts You Must Know

A pair of linear equations in two variables:

a_1 x + b_1 y + c_1 = 0

a_2 x + b_2 y + c_2 = 0

Consistency conditions:

Condition	Geometric interpretation	Solution
$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$	Lines intersect	Unique solution
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Lines coincide	Infinite solutions
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$	Parallel lines	No solution

Important Methods

Express one variable in terms of the other from one equation
Substitute into the second equation
Solve the resulting single-variable equation
Back-substitute to find the other variable

Multiply equations by suitable constants to make coefficients of one variable equal
Add or subtract the equations to eliminate that variable
Solve for the remaining variable
Substitute back

\frac{x}{b_1 c_2 - b_2 c_1} = \frac{y}{c_1 a_2 - c_2 a_1} = \frac{1}{a_1 b_2 - a_2 b_1}

Useful when the answer needs to be expressed as fractions or for quick work. Most reliable when coefficients are messy.

Solved Previous Year Questions

PYQ 1 — Elimination (4 marks)

Solve: $3x + 4y = 10$ and $2x − 3y = 1$

Multiply equation 1 by 3: $9x + 12y = 30$

Multiply equation 2 by 4: $8x - 12y = 4$

Add: $17x = 34 \Rightarrow x = 2$

Substitute in equation 1: $6 + 4y = 10 \Rightarrow y = 1$

Solution: $x = 2, y = 1$

PYQ 2 — Word Problem (4 marks)

Five years ago, Naina was 3 times as old as Atul. Ten years later, Naina will be twice as old as Atul. Find their present ages.

Let Naina’s present age = $x$ , Atul’s = $y$ .

Five years ago: $x - 5 = 3(y - 5) \Rightarrow x - 3y = -10$ … (1)

Ten years later: $x + 10 = 2(y + 10) \Rightarrow x - 2y = 10$ … (2)

Subtract (1) from (2): $y = 20$

From (2): $x = 10 + 2(20) = 50$

Naina is 50 years old, Atul is 20 years old.

Verify: 5 years ago: 45 = 3(15) ✓. Ten years later: 60 = 2(30) ✓

PYQ 3 — Consistency Check (2 marks)

Check whether the pair $2x + 3y = 7$ and $4x + 6y = 14$ is consistent.

$\frac{a_1}{a_2} = \frac{2}{4} = \frac{1}{2}$ , $\frac{b_1}{b_2} = \frac{3}{6} = \frac{1}{2}$ , $\frac{c_1}{c_2} = \frac{7}{14} = \frac{1}{2}$

All ratios are equal → infinitely many solutions (consistent, dependent). The equations represent the same line.

Difficulty Distribution

Level	%	Topics
Easy	30%	Substitution with simple numbers, consistency check
Medium	40%	Elimination with fractions, standard word problems
Hard	30%	Cross-multiplication, complex word problems

Expert Strategy

For word problems: Age problems, mixture problems, and upstream-downstream problems are the most common categories. Practice each category until you can set up equations in under 30 seconds.

For elimination: Always check whether adding or subtracting equations eliminates a variable before multiplying by new coefficients. Sometimes the equations already have equal coefficients.

Verify your answer by substituting both values into BOTH original equations. If either check fails, you made an arithmetic error.

In upstream-downstream problems: let speed of boat in still water = $b$ , speed of stream = $s$ . Upstream speed = $b - s$ , downstream speed = $b + s$ . These are the two equations. Practice this template until it’s automatic.

Common Traps

Trap 1: In the elimination method, when you subtract equations, students sometimes subtract some terms and add others by mistake. Be very deliberate: write out both equations side by side and subtract term by term.

Trap 2: In the consistency ratio check, writing the ratios as $\frac{a_1}{a_2}$ but accidentally computing $\frac{a_2}{a_1}$ . Always keep the equation-1 values in the numerator.

Trap 3: In word problems, choosing the variable assignment incorrectly (e.g., assigning $x$ to the older person when the problem says “younger is x times older”). Read the problem statement once more before writing equations.

Trap 4: Forgetting to verify the solution in both equations. CBSE marking schemes often have a verification step worth 1 mark. Never skip it.