Linear Equations in Two Variables Class 9 Maths Worksheet with Answers Chapter 4

Class 9 Maths chapter 4, Linear Equations in Two Variables worksheet. In this worksheet you will find 4 levels: Basic (warm-up), Standard (concept practice), Advance (mixed reasoning), and HOTS (challenge set). Do not attempt this worksheet before finishing the chapter. When you understand the examples in the textbook. Start from the basics and move one by one. Do not jump to HOTS like a video game boss level. Below are the primary sections of the chapter.

• Equation in two variables
• Solution as ordered pairs
• Checking given solutions
• Table of values
• Word problems framing

Class 9 Maths Worksheet – Chapter 4: Linear Equations in Two Variables

Basic

1. Write each in the form ax + by + c = 0, and write values of a, b, c:
   • 2x + 3y = 9
   • x − 4 = 3y
   • 4 = 5x − 3y
   • 2x = y
2. Convert each into a linear equation in two variables (use x and y):
   • x = −5
   • y = 2
   • 3x = 6
   • 5y = −10
3. For the equation x + y = 12, write any four solutions (ordered pairs).
4. Check which of these are solutions of 2x + 3y = 12:
   • (3, 2)
   • (0, 4)
   • (1, 4)
   • (6, 0)
   • (2, 8/3)
   • (−5, 22/3)
5. Make a table of any four solutions for x − 2y = 4 by choosing values of x.
6. Word statement to equation:
   Two friends together score 176 runs. If one scores x and the other scores y, write the equation.
   Also write any two possible solutions.

Standard

1. Write four solutions for each:
   • 2x + y = 7
   • x = 4y
2. Find the value of k if (2, 1) is a solution of 2x + 3y = k.
3. Find two solutions for each:
   • 4x + 3y = 12
   • 2x + 5y = 0
   • 3y + 4 = 0
4. Check which are solutions of x − 2y = 4:
   • (0, 2)
   • (2, 0)
   • (4, 0)
   • (1, 1)
   • (−2, −3)
   • (6, 1)
5. If y = 3x + 5, decide which is true and write one-line reason:
   • Unique solution
   • Only two solutions
   • Infinitely many solutions
6. Intercepts style (no graph needed):
   For 2x + 3y = 12, find:
   • The point where it meets the x-axis (put y = 0)
   • The point where it meets the y-axis (put x = 0)

Advance

1. Find five different solutions of x + 2y = 6.
   (Include at least one fractional solution.)
2. If (a, b) is a solution of 2x + 3y = 12, check whether (a + 3, b − 2) is also a solution.
   (Answer with Yes/No and show work.)
3. Form an equation:
   The cost of a notebook is twice the cost of a pen.
   Take notebook cost = x and pen cost = y. Write the equation.
   Also write any three possible solutions.
4. Find the missing value:
   If (x, 3) is a solution of 5x − 2y = 4, find x.
   If (2, y) is a solution of the same equation, find y.
5. Create solutions smartly:
   For 3x − 2y = 8, choose x = 0, 2, 4, 6 and find corresponding y.
   Write the four ordered pairs.
6. Compare two equations:
   Are the equations 2x + 3y = 12 and 4x + 6y = 24 representing the same set of solutions?
   Write Yes/No and one-line reason.

HOTS

1. Prove (in 2–3 lines) that a linear equation in two variables has infinitely many solutions.
   (Hint: Choose any x, solve for y.)
2. A student says: “(4, 0) is a solution of 2x + 3y = 12 because 4 + 0 = 12.”
   Find the mistake and correct it.
3. Without solving fully, decide if (1, 1), (2, 2), (3, 3) can all be solutions of one linear equation.
   If yes, write one such equation. If no, write a reason.
4. If (p, q) is a solution of x − 2y = 4, show that (p + 2, q + 1) is also a solution.
5. The line x + y = 10 passes through (a, 4). Find a.
   Then write one more point on the same line with y = 7.
6. Find all solutions (x, y) of 3y + 4 = 0 where x is an integer and −2 ≤ x ≤ 2.
   List the ordered pairs.

Answer Key

Basic – Answers

1. • 2x + 3y − 9 = 0 ⇒ a=2, b=3, c=−9
   • x − 3y − 4 = 0 ⇒ a=1, b=−3, c=−4
   • 5x − 3y − 4 = 0 ⇒ a=5, b=−3, c=−4
   • 2x − y + 0 = 0 ⇒ a=2, b=−1, c=0

   Hint: Move everything to LHS.

2. • x = −5 ⇒ x + 0y + 5 = 0
   • y = 2 ⇒ 0x + y − 2 = 0
   • 3x = 6 ⇒ 3x + 0y − 6 = 0
   • 5y = −10 ⇒ 0x + 5y + 10 = 0

   Hint: Add missing variable with coefficient 0.

3. Example solutions: (0,12), (2,10), (5,7), (12,0)
   Hint: Pick x, then y = 12 − x.
4. • (3,2): Yes
   • (0,4): Yes
   • (1,4): No (2+12=14)
   • (6,0): Yes
   • (2, 8/3): Yes (4+8=12)
   • (−5, 22/3): Yes (−10+22=12)

   Hint: Substitute and check LHS = RHS.

5. Example (choose x values):
   • x=0 ⇒ 0−2y=4 ⇒ y=−2 ⇒ (0, −2)
   • x=2 ⇒ 2−2y=4 ⇒ y=−1 ⇒ (2, −1)
   • x=4 ⇒ 4−2y=4 ⇒ y=0 ⇒ (4, 0)
   • x=6 ⇒ 6−2y=4 ⇒ y=1 ⇒ (6, 1)

   Hint: Solve for y each time.

6. Equation: x + y = 176
   Example solutions: (80,96), (100,76)
   Hint: Any pair adding to 176 works.

Standard – Answers

1. 2x + y = 7 (examples): (0,7), (1,5), (2,3), (3,1)
   x = 4y (examples): (0,0), (4,1), (8,2), (12,3)
   Hint: Choose one variable, compute the other.
2. k = 2(2) + 3(1) = 7
   Hint: Substitute (2,1).
3. • 4x + 3y = 12: (0,4), (3,0)
   • 2x + 5y = 0: (0,0), (5, −2)
   • 3y + 4 = 0: y = −4/3, so (0, −4/3), (1, −4/3)

   Hint: Use x=0 or y=0 quickly.

4. For x − 2y = 4:
   • (0,2): No (0−4=−4)
   • (2,0): No (2−0=2)
   • (4,0): Yes
   • (1,1): No (1−2=−1)
   • (−2,−3): Yes (−2+6=4)
   • (6,1): Yes (6−2=4)

   Hint: Compute LHS.

5. Infinitely many solutions
   Reason: For every real x, y becomes 3x+5 giving a new solution pair.
6. • y=0 ⇒ 2x=12 ⇒ x=6 ⇒ (6,0)
   • x=0 ⇒ 3y=12 ⇒ y=4 ⇒ (0,4)

   Hint: Axis intercepts come from setting other variable 0.

Advance – Answers

1. For x + 2y = 6:
   Example solutions: (6,0), (4,1), (2,2), (0,3), (1, 5/2)
   Hint: y = (6−x)/2.
2. Yes, it is also a solution.
   Work: If 2a+3b=12, then 2(a+3)+3(b−2)=2a+6+3b−6=2a+3b=12.
3. Equation: x = 2y (or x − 2y = 0)
   Example solutions: (10,5), (6,3), (2,1)
   Hint: Take any y, double it to get x.
4. • (x,3): 5x−2(3)=4 ⇒ 5x−6=4 ⇒ x=2
   • (2,y): 5(2)−2y=4 ⇒ 10−2y=4 ⇒ y=3

   Hint: Substitute the given coordinate.

5. 3x − 2y = 8:
   • x=0 ⇒ −2y=8 ⇒ y=−4 ⇒ (0, −4)
   • x=2 ⇒ 6−2y=8 ⇒ y=−1 ⇒ (2, −1)
   • x=4 ⇒ 12−2y=8 ⇒ y=2 ⇒ (4, 2)
   • x=6 ⇒ 18−2y=8 ⇒ y=5 ⇒ (6, 5)
6. Yes
   Reason: Second equation is just the first multiplied by 2, so solutions remain the same.

HOTS – Answers

1. Let ax + by + c = 0 (b ≠ 0).
   Choose any real x, then y = (−ax − c)/b is fixed.
   So every choice of x gives a solution ⇒ infinitely many solutions.
2. Mistake: He used 2x + 3y as x + y.
   Correct check: For (4,0), LHS = 2(4)+3(0)=8 ≠ 12, so not a solution.
3. Yes.
   One equation: y = x (or x − y = 0).
   Hint: All three points lie on the same straight line.
4. Given p − 2q = 4.
   Then (p+2) − 2(q+1) = p+2 −2q−2 = p−2q = 4.
   So (p+2, q+1) is also a solution.
5. x + y = 10, point (a,4):
   a + 4 = 10 ⇒ a = 6
   With y=7: x + 7 = 10 ⇒ x = 3 ⇒ (3,7)
6. 3y + 4 = 0 ⇒ y = −4/3 for all x.
   For x = −2, −1, 0, 1, 2:
   • (−2, −4/3)
   • (−1, −4/3)
   • (0, −4/3)
   • (1, −4/3)
   • (2, −4/3)

Worksheet for Other Class 9 Maths Chapters

• Number Systems Class 9 Maths Worksheet Chapter 1
• Polynomials Class 9 Maths Worksheet Chapter 2
• Introduction to Euclid’s Geometry Class 9 Maths Worksheet Chapter 5
• Heron’s Formula Class 9 Maths Worksheet Chapter 10
• Surface Areas and Volumes Class 9 Maths Worksheet Chapter 11
• Statistics Class 9 Maths Worksheet Chapter 12