Class 9 Maths Chapter 4 Linear Equations in Two Variables NCERT Solutions: All Exercises Solved Step-by-Step

Introduction

Linear equations with two variables are a step up from what you've seen before. Think of them as ax + by + c = 0, where solutions form a straight line on a graph, not just one point. This chapter teaches you to find those solutions and plot them.

You'll practice writing equations, finding solution sets, and graphing lines. It's key for understanding functions later on. Let's break down each exercise from the NCERT book with more solved questions.

Key Formulas

The standard form is ax + by + c = 0, where a and b aren't both zero, and the highest power is 1.

To solve, choose values for x (or y) and find the matching y (or x). For graphing, pick at least two points like (0, y-intercept) and (x-intercept, 0).

Lines parallel to x-axis: y = k. Parallel to y-axis: x = k.

......................................................................................................

Exercise 4.1 Solved Examples

Question 1: Which of these are linear equations in two variables? (i) 3x + 4y = 10 (Yes)

Solution: Check degree 1 and two variables x, y. (i) Yes. (ii) 2x² + 3y = 10 No (power 2). (iii) x = 3y + 7 Yes. (iv) y = 3x + 7 No (one variable). (v) x + y = 0 Yes.

Question 2: Show 2x + 3y = 6 has infinite solutions like (0,2), (3,0).

Solution: Plug in: For (0,2): 2(0) + 3(2) = 6. Yes. (3,0): 2(3) + 3(0) = 6. Yes. Any x, y = (6 - 2x)/3 works. Infinite pairs.

Question 3: Which are linear equations in two variables? (i) 3x + 4y = 10 Yes.

Solution: Degree 1, two vars. (ii) 2x² + 3y=10 No. (iii) x=3y+7 Yes. (iv) y=3x+7 No. (v) x+y=0 Yes.

Question 4: Prove 2x + 3y = 6 infinite solutions.

Solution: (0,2): 0+6=6. (3,0):6+0=6. (-3,6): -6+18=12? Wait, y=(6-2x)/3 for x=-3: y=(6+6)/3=4. ( -3,4 ). Infinite as any x gives y.

Exercise 4.2 Solved Examples

Question 1: Give 5 solutions for x + y = 6.

Solution: x=0, y=6 → (0,6); x=1, y=5 → (1,5); x=2, y=4 → (2,4); x=3, y=3 → (3,3); x=4, y=2 → (4,2).

Question 2: 2x + y = 7.

Solution: x=0, y=7 → (0,7); x=1, y=5 → (1,5); x=2, y=3 → (2,3); x=3, y=1 → (3,1); x=4, y=-1 → (4,-1).

Question 3: 3x + 4y = 12, etc. Similar: pick x values, solve y.

Question 4: 5 solutions x+y=6: (0,6),(1,5),(2,4),(3,3),(4,2)

Solution: y=6-x.

Question 5: 2x+y=7: (0,7),(1,5),(2,3),(3,1),(4,-1)

Solution: y=7-2x.

Question 6: 3x+4y=12: (0,3),(4,0),(8,-3)

Solution: y=(12-3x)/4. x=0,y=3; x=4,y=0; x=8,y=-3.

Question 7: 4x-2y=7: (1,1/2),(3,5/2)

Solution: y=(4x-7)/2. x=1,y=(4-7)/2=-1.5/2 wait, (4-7)/2=-3/2=-1.5. Better: x=1, 4-2y=7? 4x=4, 4-2y=7 → -2y=3 → y=-3/2. x=3,12-2y=7→ -2y=-5→ y=5/2.

Exercise 4.3 Solved Examples

Question 1(i): x+y=4: (0,4),(4,0)

Solution: Plot, straight line.

1(ii): x-y=2: (0,-2),(2,0)

1(iii): y=3x: (0,0),(1,3)

1(iv): 3=2x+y: (0,3),(1.5,0) i.e. x=3/2,y=0.

Question 2: Point (-1,3) on y=x+2? 3=? -1+2=1. No.

Question 3: (2,2) on x+y=4? 2+2=4. Yes.

Question 4: Draw graph of (i) x + y = 4.

Solution: x=0, y=4 → (0,4); y=0, x=4 → (4,0). Plot points, draw line.

(ii) x - y = 2: (0,-2), (2,0).

(iii) y = 3x: (0,0), (1,3).

Question 5: Check if point in graph lies on line y = x + 2. Say (-1,3): 3 =? -1 + 2 → 3 ≠ 1. No.

Exercise 4.4 Solved Examples

Question 1: Parallel axes? (i) 4x+3y+4=0 No (both vars).

1(ii): 3y+4=0 → y=-4/3 Yes, x-axis parallel.

1(iii): y=x No.

Question 2(i): y=3 Yes x-axis.

2(ii): 2x=1 → x=1/2 Yes y-axis.

2(iii): 5x=2 No wait, yes x=2/5 parallel y.

2(iv): x+y=8 No.

Question 3: Which equations are parallel to axes? (i) 4x + 3y + 4 = 0 No.

Solution: Rewrite to standard. Parallel x-axis if no x term (y=const). Parallel y-axis if no y (x=const).

Question 4: (i) y=3 Yes (x-axis parallel).

(ii) 2x=1 Yes (y-axis).

15 FAQs

1. What is a linear equation in two variables? ax + by + c = 0 form.

2. How many solutions? Infinite, all points on the line.

3. Graph looks like? Straight line.

4. Equation for x=5? Vertical line parallel to y-axis.

5. y=0 means? Line is x-axis.

6. Solutions for x+y=5? (0,5), (1,4), (2,3), etc.

7. Graph points for 2x+y=4? (0,4), (2,0).

8. Parallel to x-axis form? y = k.

9. Verify point (2,4) on y=2x? 4=4. Yes.

10. Degree must be? 1 for linear.

11. Two points enough for line? Yes, but three confirms straight.

12. x-y=0 graph? Line y=x through origin.

13. Why infinite solutions? Every point on line satisfies.

14. x=0 line? y-axis.

15. Plot 3x+2y=6? (0,3), (2,0).

Need help practicing these or clearing doubts? Check out www.fuzymathacademy.com for live classes and personalized math coaching. We've got your back for Class 9 CBSE exams.