In the xy-plane, the graphs of the linear function 3x + y = 6 and the quadratic function y… — SAT

In the $xy$ -plane, the graphs of the linear function $3x + y = 6$ and the quadratic function $y = 3x^2 - 5x - 2$ intersect at two points. What is the sum of the $y$ -coordinates of these two intersection points?

10Answer
B
$\frac{2}{3}$
C
14
D
2

Answer

The correct answer is 10. By rewriting the linear equation as

y = 6 - 3x

and substituting it into the quadratic equation, we get

6 - 3x = 3x^2 - 5x - 2

. Collecting all terms on one side yields the quadratic equation

3x^2 - 2x - 8 = 0

. Factoring this equation gives

(3x + 4)(x - 2) = 0

, which results in

x = 2

and

x = -\frac{4}{3}

. Substituting these

x

-values back into the linear equation gives the

y

-coordinates:

y = 6 - 3(2) = 0

and

y = 6 - 3(-\frac{4}{3}) = 10

. The sum of these

y

-coordinates is

0 + 10 = 10

Step-by-Step Solution

Express

y

in terms of

x

using the linear equation.

y = 6 - 3x

Isolating

y

makes substitution into the quadratic equation straightforward.

Substitute the expression for

y

into the quadratic equation and set the equation to zero.

6 - 3x = 3x^2 - 5x - 2

, which simplifies to

3x^2 - 2x - 8 = 0

This forms a single quadratic equation in terms of

x

to find the

x

-coordinates of the intersection points.

Solve the quadratic equation by factoring.

(3x + 4)(x - 2) = 0

, yielding

x = 2

and

x = -\frac{4}{3}

Factoring determines the values of

x

at the points of intersection.

Substitute the

x

-values back into the linear equation

y = 6 - 3x

to find the corresponding

y

-coordinates.

For

x = 2

y = 6 - 3(2) = 0

. For

x = -\frac{4}{3}

y = 6 - 3(-\frac{4}{3}) = 10

This identifies the coordinates of the two intersection points as

(2, 0)

and

(-\frac{4}{3}, 10)

Calculate the sum of the

y

-coordinates.

0 + 10 = 10

The question asks for the sum of the

y

-coordinates of the two intersection points.

Key Concept

Solving a system of linear and quadratic equations using substitution.

Estimated Time:1m 30s

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