A system of equations consists of a parabola with the equation y = x^2 and a line with the… — ACT

A system of equations consists of a parabola with the equation $y = x^2$ and a line with the equation $y = 3x - 2$ . The line and the parabola intersect at two points, $(x_1, y_1)$ and $(x_2, y_2)$ . What is the product of the $y$ -coordinates, $y_1 \cdot y_2$ , of these two points?

Answer: 4

Answer

The product of the y-coordinates of the intersection points is 4.

Solving the system of equations by setting

x^2 = 3x - 2

leads to the quadratic equation

x^2 - 3x + 2 = 0

, which factors as

(x - 1)(x - 2) = 0

. The x-coordinates of the intersection points are

x = 1

and

x = 2

. Substituting these into

y = x^2

gives the corresponding y-coordinates as

y = 1^2 = 1

and

y = 2^2 = 4

. The product of these y-coordinates is

1 \cdot 4 = 4

Step-by-Step Solution

Equate the two equations to find the x-coordinates of the intersection points.

x^2 = 3x - 2

Since both equations are solved for y, setting them equal to each other allows us to find the x-coordinates where they intersect.

Move all terms to one side to set the quadratic equation to zero.

x^2 - 3x + 2 = 0

This puts the equation in the standard form

ax^2 + bx + c = 0

, which can be solved by factoring.

Factor the quadratic equation.

(x - 1)(x - 2) = 0

, so

x_1 = 1

and

x_2 = 2

Factoring shows that the values of x that satisfy the equation are the roots of the quadratic.

Substitute the x-coordinates back into the equation of the parabola to find the y-coordinates.

For

x_1 = 1

y_1 = 1^2 = 1

. For

x_2 = 2

y_2 = 2^2 = 4

Substituting x into either of the original equations gives the corresponding y-coordinates of the intersection points.

Calculate the product of the y-coordinates.

1 \cdot 4 = 4

The question asks for the product of the y-coordinates, so we multiply

y_1

and

y_2

Key Concept

Solving systems of linear and quadratic equations by substitution.

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