Class 9 · Maths

Linear Equations in Two Variables

An equation of the form ax + by + c = 0, where a and b are not both zero, is a linear equation in two variables. This chapter covers writing such equations, finding their solutions as ordered pairs, and drawing their graphs — which are always straight lines.

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Exercise 4.1

Q1. Exercise 4.1 — Question 1

The cost of a pen is ₹5 more than the cost of a pencil. Write this as a linear equation in two variables.

Let the cost of a pencil be ₹y and the cost of a pen be ₹x. The statement says the pen costs ₹5 more than the pencil: x = y + 5 Bringing every term to one side gives the standard form: x − y − 5 = 0

Formula: Standard form: ax + by + c = 0

Final Answer: x − y − 5 = 0, where x = cost of a pen and y = cost of a pencil.

Common Mistake: Writing y = x + 5, which reverses the meaning and makes the pencil dearer.

Exam Tip: Always state what your variables represent — it is worth a mark on its own.

Exercise 4.2

Q1. Exercise 4.2 — Question 1

Verify that x = 2, y = 1 is a solution of 2x + 3y = 7, and explain why this equation has infinitely many solutions.

Substitute x = 2 and y = 1 into the left-hand side: LHS = 2(2) + 3(1) = 4 + 3 = 7 = RHS Since LHS = RHS, the pair (2, 1) satisfies the equation. For the second part: we may choose any value of x, substitute it, and solve for the matching y. Because x can take infinitely many values and each one gives a corresponding y, the equation has infinitely many solutions — geometrically, every point on its straight-line graph is a solution.

Final Answer: (2, 1) is a solution; the equation has infinitely many solutions because each chosen x gives a matching y.

Common Mistake: Finding one solution and concluding it is the only one.

Exam Tip: A linear equation in TWO variables never has a unique solution on its own.

Exercise 4.3

Q1. Exercise 4.3 — Question 1

Find three solutions of x + 2y = 6 that you could use to draw its graph.

Choose convenient values of x and solve for y. If x = 0: 0 + 2y = 6 ⇒ y = 3 ⇒ (0, 3) If x = 2: 2 + 2y = 6 ⇒ 2y = 4 ⇒ y = 2 ⇒ (2, 2) If x = 6: 6 + 2y = 6 ⇒ 2y = 0 ⇒ y = 0 ⇒ (6, 0) Plotting these three points and joining them gives the straight-line graph of the equation.

Final Answer: (0, 3), (2, 2) and (6, 0)

Common Mistake: Plotting only two points, so an arithmetic slip cannot be spotted.

Exam Tip: Pick x = 0 and y = 0 first — they give the axis intercepts and are easiest to plot.

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