The table below shows several values of x and the corresponding values of two linear functions, f and g. | x | f(x) | g(x) | | :---: | :---: | :---: | | -2 | 12 | -3 | | 0 | 6 | 1 | | 2 | 0 | 5 | | 4 | -6 | 9 | If the system of equations y = f(x) and y = g(x) has solution (x, y), what is the value o

The table below shows several values of x and the corresponding values of two linear funct… — SAT

The table below shows several values of $x$ and the corresponding values of two linear functions, $f$ and $g$ .

$x$	$f(x)$	$g(x)$
$-2$	$12$	$-3$
$0$	$6$	$1$
$2$	$0$	$5$
$4$	$-6$	$9$

If the system of equations $y = f(x)$ and $y = g(x)$ has solution $(x, y)$ , what is the value of $x + y$ ?

A
$-20$
$4$ Answer
C
$10$
D
$\frac{16}{7}$

Answer

The sum of the coordinates of the solution to the system is

4

To find the solution to the system

y = f(x)

and

y = g(x)

, we must first determine the equations of the linear functions

f

and

g

from the given table. For

f(x)

, using the points

(0, 6)

and

(2, 0)

, the slope is

\frac{0 - 6}{2 - 0} = -3

. Since the

y

-intercept is

6

, the equation is

f(x) = -3x + 6

. For

g(x)

, using the points

(0, 1)

and

(2, 5)

, the slope is

\frac{5 - 1}{2 - 0} = 2

. Since the

y

-intercept is

1

, the equation is

g(x) = 2x + 1

. Setting the two equations equal to find their intersection gives

-3x + 6 = 2x + 1

. Solving for

x

yields

5x = 5

, or

x = 1

. Substituting

x = 1

into

g(x)

gives

y = 2(1) + 1 = 3

. The sum of the coordinates of the solution is

x + y = 1 + 3 = 4

Step-by-Step Solution

Determine the linear equation for

f(x)

using the table values.

f(x) = -3x + 6

The slope of

f

is calculated as

\frac{f(2) - f(0)}{2 - 0} = \frac{0 - 6}{2} = -3

. Since

f(0) = 6

, the

y

-intercept is

6

Determine the linear equation for

g(x)

using the table values.

g(x) = 2x + 1

The slope of

g

is calculated as

\frac{g(2) - g(0)}{2 - 0} = \frac{5 - 1}{2} = 2

. Since

g(0) = 1

, the

y

-intercept is

1

Set

f(x) = g(x)

to solve for the

x

-coordinate of the intersection.

x = 1

Equating the two expressions gives

-3x + 6 = 2x + 1

. Adding

3x

to both sides and subtracting

1

from both sides results in

5x = 5

, which simplifies to

x = 1

Substitute

x = 1

back into either equation to solve for

y

y = 3

Using the equation

g(x) = 2x + 1

, substituting

x = 1

gives

y = 2(1) + 1 = 3

Calculate the sum

x + y

4

Adding the coordinates of the solution

(1, 3)

gives

1 + 3 = 4

Key Concept

Solving a system of linear equations derived from a table of values.

Question

Answer

Step-by-Step Solution

Key Concept

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