A system of equations consists of the linear equation y = x + 5 and the quadratic equation… — ACT

A system of equations consists of the linear equation $y = x + 5$ and the quadratic equation $y = (x - 3)^2 + 2$ . If the graphs of these two equations intersect at two distinct points, $(x_1, y_1)$ and $(x_2, y_2)$ , in the standard $(x, y)$ coordinate plane, what is the value of $y_1 + y_2$ ?

A
3
B
7
C
11
D
15
17Answer

Answer

To find the points of intersection, we set the two equations equal:

x + 5 = (x - 3)^2 + 2

. Expanding the right side results in

x + 5 = x^2 - 6x + 9 + 2

, which simplifies to

x + 5 = x^2 - 6x + 11

. Subtracting

x + 5

from both sides gives the quadratic equation

x^2 - 7x + 6 = 0

. Factoring this equation yields

(x - 1)(x - 6) = 0

, giving

x

-coordinates of

1

and

6

. Substituting these values into the linear equation

y = x + 5

gives the

y

-coordinates:

y = 1 + 5 = 6

and

y = 6 + 5 = 11

. The sum of these

y

-coordinates is

6 + 11 = 17

Step-by-Step Solution

Expand the quadratic equation to its standard polynomial form.

y = x^2 - 6x + 11

Expanding

(x - 3)^2 + 2

allows for easier substitution and algebraic manipulation.

Equate the linear and quadratic expressions for

y

to find the intersection points.

x + 5 = x^2 - 6x + 11

Setting the two equations equal to each other identifies the

x

-coordinates where the graphs intersect.

Rearrange the equation into standard quadratic form and solve for

x

by factoring.

x^2 - 7x + 6 = 0 \implies (x - 1)(x - 6) = 0

, which gives

x = 1

and

x = 6

Factoring allows us to find the roots of the quadratic equation representing the intersection

x

-coordinates.

Substitute the

x

-coordinates back into the linear equation

y = x + 5

to determine the corresponding

y

-coordinates.

For

x = 1

y = 6

. For

x = 6

y = 11

Finding the

y

-coordinates corresponding to the intersection points.

Calculate the sum of the two

y

-coordinates.

6 + 11 = 17

The question asks specifically for the value of

y_1 + y_2

Key Concept

Solving systems of linear and quadratic equations by substitution and factoring

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

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