Chapter 3: Pair of Linear Equations in Two Variables

Mathematics · CBSE Class 10 · NCERT Solutions

📖 Chapter 3 — Pair of Linear Equations in Two Variables (CBSE Class 10 Maths): Detailed notes — graphical & algebraic methods, and conditions for consistency.

Chapter 3 — Pair of Linear Equations in Two Variables

🔢 1. General Form

a₁x + b₁y + c₁ = 0 and a₂x + b₂y + c₂ = 0 form a pair of linear equations in x and y. Each equation is a straight line.

📈 2. Graphical Method — 3 Cases

Lines	Ratios	Solution	Consistency
Intersecting	a₁/a₂ ≠ b₁/b₂	Unique (one point)	Consistent
Coincident	a₁/a₂ = b₁/b₂ = c₁/c₂	Infinitely many	Consistent (dependent)
Parallel	a₁/a₂ = b₁/b₂ ≠ c₁/c₂	No solution	Inconsistent

✍️ 3. Algebraic Methods

Substitution: express one variable from one equation and substitute into the other.
Elimination: make the coefficient of one variable equal (multiply), then add/subtract to eliminate it.

💡 4. Steps for Word Problems

Assign variables → form two equations from the conditions → solve by substitution/elimination → check the answer.

🧠 Remember: a₁/a₂ ≠ b₁/b₂ → unique (intersecting). =b ratio but ≠c ratio → parallel, no solution. All three equal → coincident, infinite. Substitution & Elimination give the same answer.

⭐ Exam Tips: ① Compare ratios to tell consistency. ② Solve by substitution & elimination. ③ Graphical solution (plot both lines). ④ Word problems (age, fraction, cost, speed). ⑤ Find k for which a pair is consistent/inconsistent.

📘 EXERCISE 3.1 (Graphical & Consistency)

1Form the pair of linear equations and solve: 10 students took part; girls are 4 more than boys.

Let boys = x, girls = y. Total students: x + y = 10 …(1). Girls 4 more than boys: y = x + 4 …(2). Substitute (2) into (1): x + (x + 4) = 10 → 2x + 4 = 10 → 2x = 6 → x = 3. Then y = x + 4 = 3 + 4 = 7. ∴ 3 boys and 7 girls (lines meet at (3, 7)). [Pencils/pens: 5x+7y=50, 7x+5y=46 → x=3, y=5]

2Compare the ratios a₁/a₂, b₁/b₂, c₁/c₂ to decide intersecting / parallel / coincident.

Rule: a₁/a₂ ≠ b₁/b₂ → intersecting (unique); all three equal → coincident (infinite); a₁/a₂ = b₁/b₂ ≠ c₁/c₂ → parallel (none). (i) 5x−4y+8=0, 7x+6y−9=0 → 5/7 ≠ −4/6 → intersecting. (ii) 9x+3y+12=0, 18x+6y+24=0 → 9/18 = 3/6 = 12/24 = ½ → coincident. (iii) 6x−3y+10=0, 2x−y+9=0 → 6/2 = −3/−1 = 3 but ≠ 10/9 → parallel. ∴ intersecting, coincident, parallel respectively.

📗 EXERCISE 3.2 (Substitution)

1Solve by substitution: x + y = 14 and x − y = 4.

From the first equation: y = 14 − x. Substitute into the second: x − (14 − x) = 4 → 2x − 14 = 4 → 2x = 18 → x = 9. Then y = 14 − 9 = 5. ∴ x = 9, y = 5. [others: s=9,t=6 ; (iii) infinite ; (iv) x=2,y=3 ; (v) x=0,y=0 ; (vi) x=2,y=3]

2Solve 2x + 3y = 11 and 2x − 4y = −24; then find m for which y = mx + 3.

Subtract the two equations: (2x+3y) − (2x−4y) = 11 − (−24) → 7y = 35 → y = 5. Put y = 5 in 2x + 3y = 11 → 2x + 15 = 11 → 2x = −4 → x = −2. Now use y = mx + 3 with (x,y) = (−2, 5): 5 = m(−2) + 3 → 2 = −2m → m = −1. ∴ x = −2, y = 5, m = −1.

📙 EXERCISE 3.3 (Elimination)

1Solve by elimination: 3x + 4y = 10 and 2x − 2y = 2.

Make the y-coefficients equal. Multiply the 2nd equation by 2: 4x − 4y = 4. Add to the 1st (3x+4y=10): 7x = 14 → x = 2. Put x = 2 in 2x − 2y = 2 → 4 − 2y = 2 → 2y = 2 → y = 1. ∴ x = 2, y = 1. [others: x+y=5,2x−3y=4 → 19/5, 6/5 ; 3x−5y=4,9x−2y=7 → 9/13, −5/13]

Extra practice (4× the textbook) — MCQ, short answer & long answer.

🎯 Extra MCQ

1For what value of k do kx + 2y = 5 and 3x + y = 1 have a unique solution?

Need k/3 ≠ 2/1 → k ≠ 6. (Any k except 6.)

2The pair x+2y=3, 2x+4y=6 has —

Infinitely many solutions (coincident).

3Two lines with the same slope but different intercepts are —

Parallel (no solution).

4The solution of x=0, y=0 lies at the —

Origin.

📝 Short Answer

5Solve by elimination: 2x+y=8, x−y=1.

Add: 3x=9 → x=3; y=8−6=2. So x=3, y=2.

6Solve by substitution: x−2y=0, 3x+4y=20.

x=2y → 3(2y)+4y=20 → 10y=20 → y=2, x=4.

7For what value of k are 2x+3y=4 and (k+2)x+6y=3k+2 coincident?

Need 2/(k+2)=3/6=4/(3k+2). From 2/(k+2)=1/2 → k+2=4 → k=2; check 4/(3×2+2)=4/8=1/2 ✓. So k=2.

8State the condition for a pair of equations to be inconsistent.

a₁/a₂ = b₁/b₂ ≠ c₁/c₂ (parallel lines, no solution).

9The sum of two numbers is 30 and their difference is 8. Find them.

x+y=30, x−y=8 → x=19, y=11.

📖 Long Answer

10The cost of 2 kg apples and 1 kg grapes is ₹160; 4 kg apples and 2 kg grapes cost ₹300. Is this pair consistent? Interpret.

2a+g=160, 4a+2g=300 → ratios 2/4=½, 1/2=½, 160/300≠½ → a₁/a₂=b₁/b₂≠c₁/c₂ → inconsistent (no solution). The data is contradictory.

11A fraction becomes 9/11 if 2 is added to both numerator and denominator, and 5/6 if 3 is added to both. Find the fraction.

(x+2)/(y+2)=9/11 and (x+3)/(y+3)=5/6. Solving: 11x−9y=−4 and 6x−5y=−3 → x=7, y=9 → fraction 7/9.

12Solve graphically: x+y=4 and 2x−y=2. Give the solution.

Adding: 3x=6 → x=2, y=2. The lines intersect at (2,2) — unique solution, consistent.

13Father is 3 times as old as his son. After 12 years he will be twice as old. Find their ages.

f=3s and f+12=2(s+12) → 3s+12=2s+24 → s=12, f=36.

14For what value of k does kx − y = 2 and 6x − 2y = 3 have no solution?

Need k/6 = −1/−2 ≠ 2/3 → k/6=1/2 → k=3 (and 2/3 ≠ 1/2 holds). So k=3.

15The larger of two supplementary angles exceeds the smaller by 18°. Find them.

x+y=180, x−y=18 → x=99°, y=81°.