Find the equation of the pair of lines represented by the equation x^2 - 4y^2 = 0.
Correct Answer: x = 2y and x = -2y
- Step 1: Start with the given equation: x^2 - 4y^2 = 0.
- Step 2: Recognize that this equation is a difference of squares, which can be factored.
- Step 3: Use the difference of squares formula: a^2 - b^2 = (a - b)(a + b). Here, a = x and b = 2y.
- Step 4: Apply the formula to factor the equation: (x - 2y)(x + 2y) = 0.
- Step 5: Set each factor equal to zero to find the lines: x - 2y = 0 and x + 2y = 0.
- Step 6: Solve each equation for y: From x - 2y = 0, we get y = x/2; From x + 2y = 0, we get y = -x/2.
- Step 7: The equations of the lines are y = x/2 and y = -x/2.
- Factoring Quadratic Equations – Understanding how to factor a quadratic equation to find its roots, which can represent lines in this case.
- Identifying Linear Equations – Recognizing that the factored form leads to linear equations that represent pairs of lines.
