CBSE X Maths — Pair of Linear Equations in Two Variables: Notes, Practice Questions & Worked Solutions

CBSE Class X Mathematics (041/241) · Unit II: Algebra

Pair of Linear Equations in Two Variables

Two linear equations together describe two straight lines. Whether they cross, run parallel, or lie on top of each other tells you exactly how many solutions the pair has — before you even solve them.

1. Consistency — How Many Solutions?

For a pair a₁x+b₁y=c₁ and a₂x+b₂y=c₂:

Condition	Lines	Solutions
a₁/a₂ ≠ b₁/b₂	Intersecting	Exactly one (unique)
a₁/a₂ = b₁/b₂ = c₁/c₂	Coincident	Infinitely many
a₁/a₂ = b₁/b₂ ≠ c₁/c₂	Parallel	None

Worked example — checking consistency Check the pair x + 2y = 4 and 2x + 4y = 12.
a₁/a₂ = 1/2, b₁/b₂ = 2/4 = 1/2, c₁/c₂ = 4/12 = 1/3.
Since a₁/a₂ = b₁/b₂ ≠ c₁/c₂, the lines are parallel — no solution.

2. Solving Algebraically

Substitution: solve one equation for one variable, substitute into the other. Elimination: multiply the equations so a variable’s coefficients match, then add or subtract to eliminate it.

Worked example — substitution Solve x + y = 14 and x − y = 4.
From the first equation: x = 14 − y. Substitute into the second: (14 − y) − y = 4 → 14 − 2y = 4 → y = 5.
Then x = 14 − 5 = 9. Solution: x = 9, y = 5.

Worked example — elimination Solve 3x + 2y = 11 and 2x + 3y = 4.
Multiply the first by 3 and the second by 2: 9x + 6y = 33 and 4x + 6y = 8.
Subtract: 5x = 25 → x = 5. Substitute into the first equation: 3(5) + 2y = 11 → 2y = −4 → y = −2.
Solution: x = 5, y = −2.

Worked example — word problem 2 pens and 3 pencils cost ₹17. 4 pens and 1 pencil cost ₹19. Find the cost of each.
Let a pen cost x and a pencil cost y: 2x + 3y = 17 …(1) and 4x + y = 19 …(2).
From (2): y = 19 − 4x. Substitute into (1): 2x + 3(19 − 4x) = 17 → 2x + 57 − 12x = 17 → −10x = −40 → x = 4.
Then y = 19 − 16 = 3. A pen costs ₹4, a pencil costs ₹3.

Common mistakes to watch for

Comparing only a₁/a₂ and b₁/b₂ and forgetting to check c₁/c₂ when deciding between “coincident” and “parallel.”
Sign errors when multiplying an entire equation by a negative number during elimination.
Forgetting to substitute the found value back in to solve for the second variable.

Key Vocabulary Recap

Consistent — a pair of equations that has at least one solution

Inconsistent — a pair of equations with no solution

Coincident lines — the same line, giving infinitely many solutions

Syllabus Reference

This note covers CBSE Class X Mathematics Unit II, Pair of Linear Equations in Two Variables: graphical interpretation, consistency/inconsistency, algebraic conditions for the number of solutions, and solving by substitution and elimination, including simple situational problems.

Practice questions

CBSE Class X Mathematics · Unit II: Pair of Linear Equations — Practice Question Bank (no answers)

Pair of Linear Equations — Practice

Covers: consistency conditions, substitution, elimination, and situational problems. 10 questions.

Solve by substitution: x + y = 14, x − y = 4.

Solve by elimination: 2x + 3y = 12, x − y = 1.

Check whether the pair 2x + 3y = 9 and 4x + 6y = 18 is consistent, and describe the type of lines they represent.

Check whether the pair x + 2y = 4 and 2x + 4y = 12 has a solution. Justify your answer using the ratio of coefficients.

Solve algebraically and verify: x − y = 1, 2x + y = 8.

The sum of two numbers is 8 and their difference is 2. Find the numbers.

2 pens and 3 pencils cost ₹17. 4 pens and 1 pencil cost ₹19. Find the cost of a pen and a pencil.

For what value of k does the pair kx + 2y = 5, 3x + y = 1 have a unique solution?

Find the value of k for which the system 2x + ky = 1, 3x − 5y = 7 has no solution.

Q10

Solve using elimination: 3x + 2y = 11, 2x + 3y = 4.

More Practice

Q11

Solve algebraically and describe how you would verify it graphically: 2x − y = 2, 4x − y = 8.

Q12

A father’s age is 3 times his son’s age. 5 years later it will be 2.5 times the son’s age. Find their present ages.

Q13

The sum of a two-digit number and the number obtained by reversing its digits is 66. If the digits differ by 2, find the number(s).

Q14

Solve: x/2 + 2y/3 = −1, x − y/3 = 3.

Q15

Determine, using algebra, whether the pair x − 2y = 0, 3x + 4y = 20 is consistent, and find the solution if it exists.

Q16

A boat goes 30 km upstream and 44 km downstream in 10 hours. In 13 hours it can go 40 km upstream and 55 km downstream. Find the speed of the boat in still water and the speed of the stream.

Q17

State whether the pair 3x − y − 5 = 0, 6x − 2y − 10 = 0 has a unique solution, no solution, or infinitely many solutions.

Q18

The cost of 4 chairs and 3 tables is ₹2100, and the cost of 5 chairs and 2 tables is ₹1750. Find the cost of one chair and one table.

Q19

Find the value of k for which the equations x + 2y = 3, 5x + ky + 7 = 0 have no solution.

Q20

Two straight paths are represented by x − 3y = 2 and −2x + 6y = 5. State whether the paths cross, are parallel, or coincide.

Worked solutions

Show worked solutions

CBSE Class X Mathematics · Unit II: Pair of Linear Equations — Solutions

Pair of Linear Equations — Practice: Solutions

x+y=14, x−y=4.

From eq.1: x=14−y. Substitute into eq.2

(14−y)−y=4 → 14−2y=4

Solve for y, then x

y=5, x=14−5=9

✓x = 9, y = 5

2x+3y=12, x−y=1.

From eq.2: x=1+y. Substitute into eq.1

2(1+y)+3y=12 → 5y=10

✓y = 2, x = 3

2x+3y=9, 4x+6y=18 — consistency?

Compare ratios

2/4=3/6=9/18=1/2 (all equal)

✓Coincident lines — infinitely many solutions (the second equation is just the first doubled)

x+2y=4, 2x+4y=12 — solution?

Compare ratios

1/2 = 2/4 = 1/2, but 4/12=1/3 ≠ 1/2

✓a₁/a₂=b₁/b₂≠c₁/c₂ — the lines are parallel, so there is no solution

x−y=1, 2x+y=8.

From eq.1: x=1+y. Substitute into eq.2

2(1+y)+y=8 → 3y=6

✓y = 2, x = 3

Sum 8, difference 2. Find the numbers.

x+y=8, x−y=2. Add the equations

2x=10 → x=5, y=3

✓The numbers are 5 and 3

2 pens+3 pencils=₹17; 4 pens+1 pencil=₹19.

Let pen=x, pencil=y: 2x+3y=17, 4x+y=19. From eq.2: y=19−4x

2x+3(19−4x)=17 → −10x=−40

✓Pen = ₹4, Pencil = ₹3

kx+2y=5, 3x+y=1 — unique solution condition.

Need a₁/a₂ ≠ b₁/b₂

k/3 ≠ 2/1

✓k ≠ 6

2x+ky=1, 3x−5y=7 — no solution condition.

Need a₁/a₂ = b₁/b₂ ≠ c₁/c₂

2/3 = k/(−5)

Solve for k (and note c₁/c₂=1/7≠2/3, so this k does give no solution)

k = −10/3

✓k = −10/3

Q10

3x+2y=11, 2x+3y=4.

Multiply eq.1 by 3, eq.2 by 2

9x+6y=33; 4x+6y=8

Subtract to eliminate y

5x=25 → x=5

✓x = 5, y = −2

More Practice — Solutions

Q11

2x−y=2, 4x−y=8.

Subtract eq.1 from eq.2

2x=6 → x=3

✓x = 3, y = 4. Graphically, these two lines would be drawn and shown to cross exactly at (3,4).

Q12

Father=3×son; in 5 yrs father=2.5×son.

Let son=x, father=3x. Set up the second condition

3x+5=2.5(x+5) → 0.5x=7.5

✓Son = 15 years, Father = 45 years

Q13

Two-digit number + reverse = 66, digits differ by 2.

Let digits be x (tens), y (units): 11(x+y)=66 → x+y=6, and x−y=±2

If x−y=2: x=4,y=2. If x−y=−2: x=2,y=4

✓The number is 42 or 24

Q14

x/2+2y/3=−1, x−y/3=3.

Multiply eq.1 by 6, eq.2 by 3, to clear fractions

3x+4y=−6; 3x−y=9

Subtract to eliminate x

5y=−15 → y=−3, x=2

✓x = 2, y = −3

Q15

x−2y=0, 3x+4y=20.

From eq.1: x=2y. Substitute into eq.2

3(2y)+4y=20 → 10y=20

✓y=2, x=4 — a unique solution exists, so the pair is consistent (intersecting lines)

Q16

Boat/stream problem.

Let x=1/(boat−stream), y=1/(boat+stream): 30x+44y=10, 40x+55y=13. Eliminate x

Multiply by 4 and 3: 120x+176y=40; 120x+165y=39 → 11y=1 → y=1/11

Substitute back: 30x+44/11=10 → 30x=6 → x=1/5. So boat−stream=5, boat+stream=11

boat=(5+11)/2=8, stream=(11−5)/2=3

✓Boat speed = 8 km/h, Stream speed = 3 km/h

Q17

3x−y−5=0, 6x−2y−10=0.

Compare ratios

3/6=1/2, −1/−2=1/2, −5/−10=1/2 (all equal)

✓Infinitely many solutions (coincident lines — the second equation is the first doubled)

Q18

4 chairs+3 tables=2100; 5 chairs+2 tables=1750.

Let chair=c, table=t. Multiply first by 2, second by 3

8c+6t=4200; 15c+6t=5250

Subtract to eliminate t

7c=1050 → c=150; t=(2100−600)/3=500

✓Chair = ₹150, Table = ₹500

Q19

x+2y=3, 5x+ky+7=0 — no solution.

Rewrite as 5x+ky=−7. Need a₁/a₂=b₁/b₂

1/5=2/k

✓k = 10 (and c₁/c₂=3/−7≠1/5, confirming no solution)

Q20

x−3y=2, −2x+6y=5.

Compare ratios

a₁/a₂=1/−2, b₁/b₂=−3/6=−1/2 (equal); c₁/c₂=2/5

✓Since a₁/a₂=b₁/b₂≠c₁/c₂, the paths are parallel — they never cross

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