Linear Equations in Two Variables (दोन चलांतील रेषीय समीकरणे) MCQ Test — Class 9 Maharashtra SSC

Solution:

A linear equation in two variables can be written in the form ax + by + c = 0, where a, b, and c are real numbers, and a and b are not both zero.

The degree of each variable (x and y) must be 1. Option A, 2x + 3y = 5 (or 2x + 3y - 5 = 0), fits this definition as both x and y have a degree of 1.

Options B (x² + y = 7) has x with degree 2, option C (xy = 4) has a product of variables making it non-linear, and option D (3x = 9) is a linear equation in only one variable.

Solution:

The given equation is 3x = 5y - 8.

To express it in the standard form ax + by + c = 0, we need to move all terms to the left side of the equation.

Subtract 5y and add 8 to both sides: 3x - 5y + 8 = 0.

Comparing this with ax + by + c = 0, we find a = 3, b = -5, and c = 8.

Solution:

A point (x, y) is a solution to an equation if, when substituted into the equation, it makes the equation true.

Let's check option A: (1, 4) => 2(1) + 4 = 2 + 4 = 6 ≠ 7.

Let's check option B: (3, 2) => 2(3) + 2 = 6 + 2 = 8 ≠ 7.

Let's check option C: (2, 3) => 2(2) + 3 = 4 + 3 = 7. This satisfies the equation.

Let's check option D: (0, 7) => 2(0) + 7 = 0 + 7 = 7. This also satisfies the equation. Oh wait, I need to pick only ONE correct answer. I will adjust the options to ensure only one is correct. Let me re-evaluate.

Solution:

A point (x, y) is a solution to an equation if, when substituted into the equation, it makes the equation true.

For option A (1, 4): 2(1) + 4 = 2 + 4 = 6 ≠ 7.

For option B (3, 2): 2(3) + 2 = 6 + 2 = 8 ≠ 7.

For option C (2, 3): 2(2) + 3 = 4 + 3 = 7. This is true, so (2, 3) is a solution.

For option D (4, -1): 2(4) + (-1) = 8 - 1 = 7. This is also true. Oh no, I made the same mistake again. I need to make sure only ONE option is correct. Let me adjust the options once more.

Solution:

A point (x, y) is a solution to an equation if, when substituted into the equation, it makes the equation true.

For option A (1, 4): 2(1) + 4 = 2 + 4 = 6 ≠ 7.

For option B (3, 2): 2(3) + 2 = 6 + 2 = 8 ≠ 7.

For option C (2, 3): 2(2) + 3 = 4 + 3 = 7. This is true, so (2, 3) is a solution.

For option D (4, 0): 2(4) + 0 = 8 + 0 = 8 ≠ 7.

Solution:

Given the equation 3x - y = 8 and a solution (k, 2k).

Substitute x = k and y = 2k into the equation: 3(k) - (2k) = 8.

Simplify the equation: 3k - 2k = 8.

This simplifies to k = 8. So, the value of k is 8.

Solution:

A linear equation in two variables represents a straight line in a two-dimensional coordinate system.

Every point on this line is a solution to the equation.

Since a line extends infinitely in both directions and consists of an infinite number of points, a linear equation in two variables has infinitely many solutions.

Solution:

Let the cost of the storybook be x and the cost of the pen be y.

'Twice the cost of a pen' can be written as 2y.

'₹5 more than twice the cost of a pen' means we add 5 to 2y, so it becomes 2y + 5.

The cost of the storybook (x) is equal to this amount: x = 2y + 5.

Solution:

The given point is (2, -1), which means x = 2 and y = -1.

The equation is 3x + 2y = 4.

Ravi substituted y as 1 instead of -1. He wrote 2(1) instead of 2(-1).

The correct substitution would be 3(2) + 2(-1) = 6 - 2 = 4. Since 4 = 4, the point (2, -1) IS actually a solution.

Solution:

When a point (x, y) lies on the graph of an equation, it means that the coordinates of that point satisfy the equation.

Given the point (a, b) and the equation 5x - y = 12.

Substitute x = a and y = b into the equation.

This gives us 5(a) - (b) = 12, or 5a - b = 12. This statement must be true.