A system of equations consists of the quadratic function f(x) = -x^2 + 6x - 2 and the line… — SAT

A system of equations consists of the quadratic function $f(x) = -x^2 + 6x - 2$ and the linear function $g(x) = 2x + 1$ . If the graphs of $f$ and $g$ intersect at the points $(x_1, y_1)$ and $(x_2, y_2)$ in the $xy$ -plane, what is the value of $y_1 + y_2$ ?

Answer: 10

Answer

By setting the two equations equal to each other, we obtain

-x^2 + 6x - 2 = 2x + 1

. Rearranging terms to one side gives

x^2 - 4x + 3 = 0

, which factors as

(x-1)(x-3) = 0

. The x-coordinates of the intersection points are

x = 1

and

x = 3

. Substituting these into the linear equation gives the y-coordinates: for

x = 1

y = 2(1) + 1 = 3

; for

x = 3

y = 2(3) + 1 = 7

. Summing these y-coordinates gives

3 + 7 = 10

Step-by-Step Solution

Set the quadratic function equal to the linear function to find the x-coordinates of the intersection points.

-x^2 + 6x - 2 = 2x + 1

At the points of intersection, the y-values of both functions must be equal.

Rearrange the equation into standard quadratic form and solve for

x

by factoring.

x^2 - 4x + 3 = 0 \Rightarrow (x - 1)(x - 3) = 0 \Rightarrow x = 1 \text{ or } x = 3

Factoring the quadratic equation gives the x-coordinates of the intersection points.

Substitute the x-coordinates back into the linear equation to find the corresponding y-coordinates.

For

x_1 = 1

y_1 = 2(1) + 1 = 3

. For

x_2 = 3

y_2 = 2(3) + 1 = 7

The linear equation

g(x) = 2x + 1

is simpler to use for finding the y-coordinates than the quadratic equation.

Sum the two y-coordinates.

y_1 + y_2 = 3 + 7 = 10

The question asks for the sum of the y-coordinates of the intersection points.

Key Concept

Solving a system consisting of a linear equation and a quadratic equation by substitution or equating them.

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