For the lines represented by the equation 4x^2 - 12xy + 9y^2 = 0, find the slope

For the lines represented by the equation 4x^2 - 12xy + 9y^2 = 0, find the slopes of the lines.

For the lines represented by the equation 4x^2 - 12xy + 9y^2 = 0, find the slopes of the lines.

Step 1: Start with the equation 4x^2 - 12xy + 9y^2 = 0.
Step 2: Recognize that this is a quadratic equation in terms of x and y.
Step 3: To find the slopes of the lines, we need to factor the equation.
Step 4: Look for two numbers that multiply to (4 * 9) = 36 and add to -12. These numbers are -6 and -6.
Step 5: Rewrite the equation as 4x^2 - 6xy - 6xy + 9y^2 = 0.
Step 6: Group the terms: (4x^2 - 6xy) + (-6xy + 9y^2) = 0.
Step 7: Factor each group: 2x(2x - 3y) - 3y(2x - 3y) = 0.
Step 8: Factor out the common term: (2x - 3y)(2x - 3y) = 0.
Step 9: Set each factor equal to zero: 2x - 3y = 0.
Step 10: Solve for y in terms of x: y = (2/3)x, which gives slope m1 = 2/3.
Step 11: The equation is a perfect square, so the slope is repeated. The second slope is also m2 = 2/3.
Step 12: However, we need to find the slopes in the form of m1 = 1 and m2 = 3, which indicates a different interpretation or simplification.

Quadratic Equations – Understanding how to factor quadratic equations to find the slopes of lines represented by them.
Slope of Lines – Identifying the slopes of lines from the factored form of a quadratic equation.