Class 9 Maths Chapter 4 Linear Equations in Two Variables – Exercise 4.2 NCERT Solutions

Introduction

Exercise 4.2 teaches you how to solve pairs of linear equations in two variables. You’ll practice both the substitution method and the elimination method, which are essential techniques for solving simultaneous equations.

Key Methods Used

• Substitution Method:

  • Solve one equation for one variable.

  • Substitute into the other equation.

  • Solve for the second variable.

• Elimination Method:

  • Multiply equations (if needed) to align coefficients.

  • Add or subtract equations to eliminate one variable.

  • Solve for the remaining variable.

Solved Questions (Step by Step)

Q1. Solve: $x+y=5$, $x-y=1$.

Solution (Elimination):

• Add equations: $2x=6⟹x=3$.

• Substitute: $3+y=5⟹y=2$.

• Solution: $(x,y)=(3,2)$.

Q2. Solve: $2x+y=11$, $x-y=1$.

Solution (Substitution):

• From second: $x=y+1$.

• Substitute: $2(y+1)+y=11⟹2y+2+y=11⟹3y=9⟹y=3$.

• Then $x=4$.

• Solution: $(x,y)=(4,3)$.

Q3. Solve: $3x+2y=12$, $x-y=2$.

Solution:

• From second: $x=y+2$.

• Substitute: $3(y+2)+2y=12⟹3y+6+2y=12⟹5y=6⟹y=\frac{6}{5}$.

• Then $x=\frac{11}{5}$.

• Solution: $(x,y)=(\frac{11}{5},\frac{6}{5})$.

Q4. Solve: $x+2y=7$, $2x-y=3$.

Solution (Elimination):

• Multiply first by 2: $2x+4y=14$.

• Subtract second: $(2x+4y)-(2x-y)=14-3⟹5y=11⟹y=\frac{11}{5}$.

• Substitute: $x+2(\frac{11}{5})=7⟹x=\frac{9}{5}$.

• Solution: $(x,y)=(\frac{9}{5},\frac{11}{5})$.

Q5. Solve: $x+y=10$, $x-y=4$.

Solution:

• Add: $2x=14⟹x=7$.

• Substitute: $7+y=10⟹y=3$.

• Solution: $(x,y)=(7,3)$.

FAQs (10 for Exercise 4.2)

1. Q: What is the substitution method? A: Solve one equation for a variable, substitute into the other, then solve.

2. Q: What is the elimination method? A: Align coefficients, add/subtract equations to eliminate one variable.

3. Q: Which method is faster? A: Depends on the equations; elimination is often quicker when coefficients align.

4. Q: Can linear equations have no solution? A: Yes, if lines are parallel.

5. Q: Can linear equations have infinite solutions? A: Yes, if lines overlap (coincident).

6. Q: What is a unique solution? A: When two lines intersect at exactly one point.

7. Q: How do you check your solution? A: Substitute values back into both equations.

8. Q: What is the graphical meaning of solving equations? A: Finding the intersection point of two lines.

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

10. Q: Why are these methods important? A: They form the basis for solving systems of equations in algebra and real‑life problems.

Conclusion

Exercise 4.2 has 15 questions that strengthen your skills in solving simultaneous equations using substitution and elimination. This prepares you for word problems and advanced algebra applications.

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