Question 1
Form and solve the quiz-participant and pencil-pen equations graphically.
Steps only
(i) b + g = 10, g − b = 4 ⇒ (b,g) = (3,7)
(ii) 5p + 7q = 50, 7p + 5q = 46 ⇒ (p,q) = (3,5)
Answer
- (i) 3 boys and 7 girls.
- (ii) Pencil ₹3 and pen ₹5.
CLASS 10 · MATHEMATICS · CHAPTER 3
Pair of Linear Equations in Two Variables
Form pairs of linear equations, identify intersecting, parallel and coincident lines, and solve by substitution or elimination.
12 worked answersAll 3 exercise sets
What you will learn
- Connect the number of solutions with two lines.
- Solve a pair by substitution and elimination.
- Translate word problems into two equations.
12 worked answers · short steps first, explanations when you need them
Exercise 3.1
Textbook pages 28–29
Question 2
Use coefficient ratios to classify the three pairs of lines as intersecting, parallel or coincident.
Steps only
(i) 5/7 ≠ −4/6: intersecting.
(ii) 9/18 = 3/6 = 12/24: coincident.
(iii) 6/2 = (−3)/(−1) ≠ 10/9: parallel.
Answer
- (i) Intersecting;
- (ii) coincident;
- (iii) parallel.
Question 3
Use coefficient ratios to decide whether each of the five equation pairs is consistent.
Steps only
(i) Unequal coefficient ratios: consistent, unique solution.
(ii) Equal coefficient ratios but unequal constants: inconsistent.
(iii) Unequal coefficient ratios: consistent, unique solution.
(iv) All ratios equal: consistent, infinitely many solutions.
(v) Unequal coefficient ratios: consistent, unique solution.
Answer
Consistent: (i), (iii), (iv), (v). Inconsistent: (ii). Part (iv) has infinitely many solutions; the other consistent pairs have one.
Question 4
Classify the four pairs as consistent or inconsistent and give the graphical solution where it exists.
Steps only
(i) Same line: consistent with infinitely many solutions.
(ii) Parallel lines: inconsistent.
(iii) Intersection (2,2): consistent.
(iv) Parallel lines: inconsistent.
Answer
- (i) Infinite solutions;
- (ii) none;
- (iii) (2,2);
- (iv) none.
Question 5
Half the perimeter of a rectangular garden is 36 m and its length is 4 m more than its width. Find its dimensions.
Steps only
l + w = 36; l − w = 4
2l = 40 ⇒ l = 20, w = 16
Answer
Length 20 m; width 16 m.
Question 6
For 2x + 3y − 8 = 0, give equations producing intersecting, parallel and coincident pairs.
Steps only
One valid set: (i) x + y − 1 = 0; (ii) 4x + 6y − 9 = 0; (iii) 4x + 6y − 16 = 0.
Answer
Examples given; many other correct equations are possible.
Question 7
Find the vertices of the triangle formed by x − y + 1 = 0, 3x + 2y − 12 = 0 and the x-axis.
Steps only
On y = 0, the lines meet at (−1,0) and (4,0).
Solving both lines gives (2,3).
- x − y + 1 = 0
- 3x + 2y − 12 = 0
A(-1, 0) · B(4, 0) · C(2, 3)
Answer
Vertices (−1,0), (4,0), (2,3).
Exercise 3.2
Textbook pages 33–34
Question 1
Solve the six pairs of equations in Question 1 by substitution.
Steps only
(i) x = 9, y = 5
(ii) s = 9, t = 6
(iii) Infinitely many solutions: both equations are equivalent.
(iv) x = 2, y = 3
(v) x = 0, y = 0
(vi) x = 2, y = 3
Answer
- (i) (9,5);
- (ii) (9,6);
- (iii) infinitely many;
- (iv) (2,3);
- (v) (0,0);
- (vi) (2,3).
Question 2
Solve 2x + 3y = 11 and 2x − 4y = −24, then find m if y = mx + 3.
Steps only
Subtract: 7y = 35 ⇒ y = 5; then x = −2
5 = −2m + 3 ⇒ m = −1
Answer
x = −2, y = 5 and m = −1.
Question 3
Form and solve the six word problems about numbers, angles, sports goods, taxi charges, a fraction and ages.
Steps only
(i) Numbers 39 and 13.
(ii) Angles 99° and 81°.
(iii) Bat ₹500; ball ₹50.
(iv) Fixed charge ₹5; ₹10/km; 25 km costs ₹255.
(v) Fraction 7/9.
(vi) Jacob 40 years; son 10 years.
Answer
39 and 13; 99° and 81°; ₹500 and ₹50; ₹5 plus ₹10/km and ₹255; 7/9; 40 and 10 years.
Exercise 3.3
Textbook pages 36–37
Question 1
Solve all four equation pairs by elimination and substitution.
Steps only
(i) x = 19/5, y = 6/5
(ii) x = 2, y = 1
(iii) x = 9/13, y = −5/13
(iv) x = 2, y = −3
Answer
- (i) (19/5,6/5);
- (ii) (2,1);
- (iii) (9/13,−5/13);
- (iv) (2,−3).
Question 2
Form and solve the five word problems about a fraction, ages, a two-digit number, banknotes and library charges.
Steps only
(i) Fraction 3/5.
(ii) Nuri 50 years; Sonu 20 years.
(iii) Number 18.
(iv) Ten ₹50 notes and fifteen ₹100 notes.
(v) Fixed charge ₹15; extra-day charge ₹3.
Answer
3/5; 50 and 20 years; 18; 10 and 15 notes; ₹15 fixed and ₹3 per extra day.