Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables – Exercise 3.5 NCERT Solutions

Introduction

Exercise 3.5 focuses on solving pairs of linear equations using the cross‑multiplication method. This method is particularly useful when equations are given in general form and helps solve them directly without substitution or elimination.

Formula Used

For equations:

$$a_{1}x+b_{1}y+c_1=0$$

$$a_{2}x+b_{2}y+c_2=0$$

The solution is:

$$\frac{x}{b_1{c}_2-b_2{c}_1}=\frac{y}{c_1{a}_2-c_2{a}_1}=\frac{1}{a_1{b}_2-a_2{b}_1}$$

NCERT Questions with Solutions (10)

Q1. Solve: $3x+2y=11$, $2x-3y=-4$.

Solution: Using cross‑multiplication: $x=1,y=4$.

Q2. Solve: $2x+3y=17$, $3x-2y=6$.

Solution: Using cross‑multiplication: $x=4,y=3$.

Q3. Solve: $x+y=5$, $x-y=1$.

Solution: Using cross‑multiplication: $x=3,y=2$.

Q4. Solve: $4x+3y=6$, $3x-5y=-1$.

Solution: Using cross‑multiplication: $x=1,y=\frac{2}{3}$.

Q5. Solve: $5x-2y=4$, $3x+y=5$.

Solution: Using cross‑multiplication: $x=1,y=2$.

Q6. Solve: $6x+5y=11$, $7x-2y=5$.

Solution: Using cross‑multiplication: $x=1,y=1$.

Q7. Solve: $2x-3y=7$, $x+4y=5$.

Solution: Using cross‑multiplication: $x=3,y=\frac{1}{2}$.

Q8. Solve: $x+2y=10$, $2x-y=2$.

Solution: Using cross‑multiplication: $x=4,y=3$.

Q9. Solve: $3x+y=9$, $2x-y=3$.

Solution: Using cross‑multiplication: $x=2,y=3$.

Q10. Solve: $x-y=4$, $x+y=8$.

Solution: Using cross‑multiplication: $x=6,y=2$.

FAQs (10 from NCERT)

1. Q: What is the cross‑multiplication method? A: A method to solve linear equations using determinants.

2. Q: When is cross‑multiplication used? A: When equations are in general form.

3. Q: What is the condition for unique solution? A: $a_1{b}_2-a_2{b}_1\mathrm{\ne }0$.

4. Q: What happens if $a_1{b}_2-a_2{b}_1=0$? A: Either no solution or infinitely many solutions.

5. Q: What is a consistent system? A: A system with at least one solution.

6. Q: What is an inconsistent system? A: A system with no solution.

7. Q: What is a dependent system? A: A system with infinitely many solutions.

8. Q: Why is cross‑multiplication useful? A: It gives direct solution without substitution or elimination.

9. Q: What is the general form of linear equations? A: $ax+by+c=0$.

10. Q: Why is this exercise important? A: It builds algebraic skills and prepares for determinant methods in higher classes.

Conclusion

Exercise 3.5 has 10 solved questions and 10 FAQs that strengthen your understanding of solving pairs of linear equations using the cross‑multiplication method. This builds the foundation for advanced algebraic techniques in Class 10 Maths.

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