For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the sum

For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the sum of the slopes?

For the lines represented by the equation 2x^2 + 3xy + y^2 = 0, what is the sum of the slopes?

Correct Answer: -3

Step 1: Identify the given equation, which is 2x^2 + 3xy + y^2 = 0.
Step 2: Recognize that this is a quadratic equation in two variables (x and y).
Step 3: The equation can be rewritten in the standard form of a conic section, which helps us find the slopes of the lines.
Step 4: Use the formula for the sum of the slopes of the lines represented by the equation ax^2 + bxy + cy^2 = 0, where the sum of the slopes is given by -b/a.
Step 5: Identify the coefficients: a = 2, b = 3, and c = 1 from the equation.
Step 6: Substitute the values into the formula: Sum of slopes = -b/a = -3/2.
Step 7: Calculate the result: The sum of the slopes is -1.5.

Quadratic Equations and Slopes – Understanding how to derive slopes from the coefficients of a quadratic equation in two variables.
Sum of Roots – Applying the relationship between the coefficients of a quadratic equation to find the sum of the slopes.