Linear Equations in Two Variables — Chapter Test | Class 9 IMO

Q1

Find the point on the graph of 3x + 4y = 24 whose x-coordinate is double its y-coordinate.

Since the x-coordinate is double the y-coordinate, let x = 2y. Substitute x = 2y into 3x + 4y = 24: 3(2y) + 4y = 24 → 6y + 4y = 24 → 10y = 24 → y = 2.4. Then x = 2(2.4) = 4.8. The point is (4.8, 2.4).

Q2

The graph of the linear equation 2x + 3y = 6 cuts the y-axis at the point:

A graph cuts the y-axis where x = 0. Substituting x = 0 in 2x + 3y = 6 gives 2(0) + 3y = 6 → 3y = 6 → y = 2. Thus, the point is (0, 2).

Q3

A group of friends goes out to a local eatery in Bengaluru. They order x plates of idli at ₹30 per plate and y cups of filter coffee at ₹15 per cup. The total bill amounts to ₹120. Which equation represents this situation?

Cost of x plates of idli = 30x. Cost of y cups of coffee = 15y. Total cost is 30x + 15y = 120.

Q4

A straight line passes through the point (a, a) and is perpendicular to the x-axis. What is its linear equation representation?

A line perpendicular to the x-axis is a vertical line. Vertical lines have equations of the form x = k. Since it passes through (a, a), the x-coordinate must always be a. Hence, the equation is x = a.

Q5

The value of k for which the linear equation kx − y = 2 has a solution where x is always 1 more than y, and passing through (3, 2), is:

The solution point given is (3, 2). Substitute x = 3 and y = 2 into the equation kx − y = 2: k(3) − 2 = 2 → 3k = 4 → k = 4/3.

Q6

If the graph of the linear equation 5x + 2y = 20 forms a triangle with the coordinate axes, find the area of this triangle.

Find intercepts: y = 0 → 5x = 20 → x = 4. x = 0 → 2y = 20 → y = 10. The right-angled triangle has a base of 4 units along the x-axis and a height of 10 units along the y-axis. Area = ½ × base × height = ½ × 4 × 10 = 20 sq units.

Q7

If x = 1, y = 2 is a solution of the equation a²x + ay = 3, then the possible values of a are:

Substitute x = 1 and y = 2 into the equation: a²(1) + a(2) = 3 → a² + 2a − 3 = 0. Solving this quadratic equation: (a + 3)(a − 1) = 0 → a = 1 or a = −3.

Q8

The cost of a notebook is twice the cost of a pen. Writing this statement as a linear equation in two variables (where x is cost of notebook and y is cost of pen) gives:

Let cost of notebook = x and cost of pen = y. According to the given statement: Cost of notebook = 2 × Cost of pen → x = 2y → x − 2y = 0.

Q9

The coordinates of a point on the line x + 2y = 8 whose y-coordinate is half its x-coordinate is:

Given y-coordinate is half of x-coordinate, so x = 2y. Substitute x = 2y into x + 2y = 8: 2y + 2y = 8 → 4y = 8 → y = 2. Then x = 2(2) = 4. The point is (4, 2).

Q10

The graph of the linear equation x = k is a line parallel to the:

The equation x = k represents a line where the x-coordinate is always k, regardless of the value of y. This forms a straight vertical line parallel to the y-axis.

Q11

The graph of ax + by + c = 0 is a straight line passing through the origin if and only if:

If a line passes through the origin (0, 0), then substituting x = 0 and y = 0 must satisfy its equation: a(0) + b(0) + c = 0 → c = 0.

Q12

Look at the pattern: Term 1 = 5, Term 2 = 9, Term 3 = 13, Term 4 = 17. If y represents the value of the term and x represents the term number, which equation represents this rule?

Test y = 4x + 1. For x = 1: y = 4(1)+1 = 5. For x = 2: y = 4(2)+1 = 9. For x = 3: y = 4(3)+1 = 13. This linear equation perfectly matches the given values.

Q13

The total weight of a box containing x identical cricket balls (each weighing 150 grams) and a container weighing 200 grams is given by y grams. Write the linear expression.

The total weight y is the sum of the combined weight of x balls (150 grams each) and the constant weight of the empty container (200 grams). Therefore, y = 150x + 200.

Q14

The linear equation 2x − 5y = 7 has:

A linear equation in two variables represents a straight line. Since a line contains infinitely many points, the equation has infinitely many solutions.

Q15

If x is a positive real number and satisfies the equation 0.2x + 0.3(20) = 0.5(x + 6), find x.

Simplify the equation: 0.2x + 6 = 0.5x + 3 → 6 − 3 = 0.5x − 0.2x → 3 = 0.3x → x = 3 / 0.3 = 10.