Class 10 Maths Chapter 3 Pair of Linear Equations in Two Variables – Exercise 3.6 NCERT Solutions

Introduction

Exercise 3.6 applies the concepts of linear equations in two variables to real‑life word problems. You’ll learn how to translate everyday situations into equations, solve them using algebraic methods, and interpret the results meaningfully.

Formula Used

• General Form of Linear Equation in Two Variables:

$$$$ax+by+c=0

• Methods to Solve:

  • Substitution Method

  • Elimination Method

  • Cross‑Multiplication Method

NCERT Questions with Solutions (10)

Q1. The sum of two numbers is 27 and their difference is 5. Find the numbers.

Solution: Equations: $x+y=27$, $x-y=5$. Add: $2x=32\Rightarrow x=16$. Then $y=11$. Numbers = 16, 11.

Q2. The sum of two numbers is 45 and one number is twice the other. Find the numbers.

Solution: Equations: $x+y=45$, $x=2y$. Substitute: $2y+y=45\Rightarrow 3y=45\Rightarrow y=15$. Then $x=30$. Numbers = 30, 15.

Q3. The sum of the digits of a two‑digit number is 9. If 27 is added to the number, digits are reversed. Find the number.

Solution: Let digits = $x,y$. Equation: $x+y=9$. Number = $10x+y$. Reversed = $10y+x$. Equation: $10x+y+27=10y+x$. Simplify: $9x-9y=-27\Rightarrow x-y=-3$. Solve: $x+y=9$, $x-y=-3$. Add: $2x=6\Rightarrow x=3$. Then $y=6$. Number = 36.

Q4. The sum of the digits of a two‑digit number is 12. If 18 is added, digits are reversed. Find the number.

Solution: Equations: $x+y=12$. $10x+y+18=10y+x$. Simplify: $9x-9y=-18\Rightarrow x-y=-2$. Solve: $x+y=12$, $x-y=-2$. Add: $2x=10\Rightarrow x=5$. Then $y=7$. Number = 57.

Q5. A fraction becomes $\frac{1}{2}$ when 1 is subtracted from numerator and $\frac{1}{3}$ when 1 is added to denominator. Find the fraction.

Solution: Let fraction = $\frac{x}{y}$. Equation 1: $\frac{x-1}{y}=\frac{1}{2}\Rightarrow 2x-2=y$. Equation 2: $\frac{x}{y+1}=\frac{1}{3}\Rightarrow 3x=y+1$. Solve: $y=2x-2$, $y=3x-1$. Equating: $2x-2=3x-1\Rightarrow x=-1$. Check: Fraction = $\frac{-1}{-4}=\frac{1}{4}$.

Q6. A fraction becomes $\frac{3}{4}$ when 1 is added to numerator and $\frac{1}{2}$ when 1 is added to denominator. Find the fraction.

Solution: Let fraction = $\frac{x}{y}$. Equation 1: $\frac{x+1}{y}=\frac{3}{4}\Rightarrow 4x+4=3y$. Equation 2: $\frac{x}{y+1}=\frac{1}{2}\Rightarrow 2x=y+1$. Solve: $3y=4x+4$, $y=2x-1$. Substitute: $3(2x-1)=4x+4\Rightarrow 6x-3=4x+4\Rightarrow 2x=7\Rightarrow x=\frac{7}{2}$. Then $y=6$. Fraction = $\frac{7}{12}$.

Q7. The sum of two numbers is 15 and sum of their reciprocals is $\frac{10}{3}$. Find the numbers.

Solution: Equations: $x+y=15$, $\frac{1}{x}+\frac{1}{y}=\frac{10}{3}$. Simplify: $\frac{x+y}{xy}=\frac{10}{3}\Rightarrow \frac{15}{xy}=\frac{10}{3}\Rightarrow xy=\frac{45}{10}=4.5$. Solve quadratic: $t^2-15t+4.5=0$. Numbers = approx roots (not exact integers).

Q8. The sum of two numbers is 27 and sum of their reciprocals is $\frac{1}{9}$. Find the numbers.

Solution: Equations: $x+y=27$, $\frac{1}{x}+\frac{1}{y}=\frac{1}{9}$. Simplify: $\frac{27}{xy}=\frac{1}{9}\Rightarrow xy=243$. Solve quadratic: $t^2-27t+243=0$. Roots: 9, 18. Numbers = 9, 18.

Q9. The sum of two numbers is 26 and difference is 2. Find the numbers.

Solution: Equations: $x+y=26$, $x-y=2$. Add: $2x=28\Rightarrow x=14$. Then $y=12$. Numbers = 14, 12.

Q10. The sum of two numbers is 20 and product is 96. Find the numbers.

Solution: Equations: $x+y=20$, $xy=96$. Quadratic: $t^2-20t+96=0$. Solve: Roots = 12, 8. Numbers = 12, 8.

FAQs (10 from NCERT)

1. Q: What is a word problem in linear equations? A: A real‑life situation expressed mathematically as equations.

2. Q: How do you form equations from word problems? A: Translate conditions into algebraic expressions.

3. Q: What is the substitution method? A: Solve one equation for a variable and substitute into the other.

4. Q: What is the elimination method? A: Eliminate one variable by adding or subtracting equations.

5. Q: What is the cross‑multiplication method? A: Solve equations in general form using determinants.

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

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

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

9. Q: Why are word problems important? A: They connect mathematics to real‑life applications.

10. **Q: Why is this exercise important?

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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