Linear Equations in Two variables

Infinitely many solutions

Step 1: A linear equation in two variables has the form ax + by + c = 0. Step 2: For any linear equation in two variables (where both coefficients are non-zero), there are infinitely many solutions. Step 3: We can choose any value for x and find the corresponding y, or vice versa. Step 4: For example, if x = 0, then y = 2; if x = 5, then y = 0; if x = 2.5, then y = 1, etc. Step 5: Therefore, 2x + 5y = 10 has infinitely many solutions.

x = 3

Step 1: Given equation is 4x - 3y = 12 and y = 0. Step 2: Substitute y = 0 into the equation: 4x - 3(0) = 12. Step 3: Simplify: 4x - 0 = 12, so 4x = 12. Step 4: Solve for x: x = 12/4 = 3. Step 5: Verify: 4(3) - 3(0) = 12 - 0 = 12 ✓. Therefore, x = 3.

2x - 3y = 0

Step 1: For a line to pass through origin (0, 0), the point (0, 0) must satisfy the equation. Step 2: Test each option by substituting x = 0 and y = 0. Step 3: For 2x - 3y = 0: 2(0) - 3(0) = 0 - 0 = 0 ✓. Step 4: For x + y = 5: 0 + 0 = 0 ≠ 5. Step 5: The other options also don't equal their right-hand side when substituting (0, 0). Therefore, 2x - 3y = 0 passes through the origin.

y = -2

Step 1: A line parallel to x-axis has the form y = k, where k is a constant. Step 2: All points on such a line have the same y-coordinate. Step 3: Since the line passes through (3, -2), all points have y-coordinate = -2. Step 4: Therefore, the equation is y = -2. Step 5: Verify: The point (3, -2) satisfies y = -2 since the y-coordinate is -2.