Linear Equations in Two Variables

Infinitely many solutions

Step 1: The given equation 5x – 3y = 15 is a linear equation in two variables x and y. Step 2: For any value of x we choose, we get a corresponding value of y. For example, x=3 gives y=0; x=0 gives y=–5; x=6 gives y=5. Step 3: Since x can take infinitely many real values, we get infinitely many pairs (x, y) that satisfy the equation. Step 4: This is a fundamental property — a linear equation in two variables always has infinitely many solutions, representing a straight line on the Cartesian plane. Common mistake: Students confuse this with linear equations in one variable, which have a unique

3x + 2y = 70; solution (10, 20)

Step 1: Cost of 3 notebooks = 3x and cost of 2 pens = 2y. Total = 70, so equation is 3x + 2y = 70. Step 2: Check option (10, 20): 3(10) + 2(20) = 30 + 40 = 70 ✓. Step 3: Check option (20, 10): 3(20) + 2(10) = 60 + 20 = 80 ≠ 70 ✗. Step 4: The equation 2x+3y=70 reverses the coefficients and does not match the problem statement. Common mistake: Students swap the coefficients of x and y or swap the values in the solution pair.

(5, 0)

Step 1: Find k using the solution (3, 1): 2(3) + k(1) = 10 → 6 + k = 10 → k = 4. Step 2: The equation becomes 2x + 4y = 10, which simplifies to x + 2y = 5. Step 3: Check (0, 2): 0 + 2(2) = 4 ≠ 5 ✗. Check (5, 0): 5 + 2(0) = 5 ✓. Check (1, 3): 1 + 6 = 7 ≠ 5 ✗. Check (2, 2): 2 + 4 = 6 ≠ 5 ✗. Step 4: (5, 0) is the only point that satisfies the equation. Common mistake: Students forget to first find k before checking other solutions.

A line parallel to the x-axis

Step 1: If a = 0, the equation becomes 0·x + by + c = 0, i.e., by + c = 0. Step 2: Solving: y = –c/b, which is a constant value (say k). Step 3: Since y = constant for all values of x, every point on this line has the same y-coordinate regardless of x. Step 4: A set of all points with the same y-coordinate forms a horizontal line, which is parallel to the x-axis. Common mistake: Students confuse 'parallel to x-axis' with 'parallel to y-axis'. When x is missing (a=0), the line is horizontal (parallel to x-axis); when y is missing (b=0), the line is vertical (parallel to y-axis).