In the xy-plane, the graph of the linear function f has a slope of m and a y-intercept of … — SAT

In the $xy$ -plane, the graph of the linear function $f$ has a slope of $m$ and a $y$ -intercept of $(0, b)$ . The graph of the linear function $g$ has a slope of $-3m$ and a $y$ -intercept of $(0, b + 15)$ . If the graphs of $f$ and $g$ intersect at the point $(3, 5)$ , what is the value of $f(5)$ ?

Answer: 7.5

Answer

7.5 (or 15/2)

By writing the equations of the two lines as

f(x) = mx + b

and

g(x) = -3mx + b + 15

, and substituting the intersection point

(3, 5)

, we get the system of equations

3m + b = 5

and

-9m + b = -10

. Subtracting these equations gives

12m = 15

, which simplifies to

m = 1.25

. Substituting

m

back into

3m + b = 5

yields

b = 1.25

. The function is therefore

f(x) = 1.25x + 1.25

. Evaluating this at

x = 5

gives

f(5) = 1.25(5) + 1.25 = 7.5

Step-by-Step Solution

Formulate equations for both linear functions using slope-intercept form.

f(x) = mx + b

and

g(x) = -3mx + b + 15

The equation of a line with slope

M

and

y

-intercept

(0, B)

is given by

y = Mx + B

Substitute the coordinates of the intersection point

(3, 5)

into both equations.

3m + b = 5

and

-9m + b = -10

Since the intersection point lies on both lines, its coordinates must satisfy both equations.

Solve the system of equations for

m

and

b

m = 1.25

and

b = 1.25

Subtracting the equations eliminates

b

, giving

12m = 15 \implies m = 1.25

. Substituting this back into the first equation gives

3(1.25) + b = 5 \implies b = 1.25

Evaluate

f(5)

using the found values.

f(5) = 1.25(5) + 1.25 = 7.5

Substitute

x = 5

into the equation

f(x) = 1.25x + 1.25

Key Concept

Solving systems of linear equations and evaluating linear functions from graph intersections.

Estimated Time:2m 0s

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