In the xy-plane, the graph of the linear function f passes through the point (6, 4). Line … — SAT

In the $xy$ -plane, the graph of the linear function $f$ passes through the point $(6, 4)$ . Line $g$ is perpendicular to the graph of $f$ and has a $y$ -intercept of $(0, 12)$ . The graph of $f$ and line $g$ intersect at the point $P(h, k)$ , where $h$ and $k$ are integers. If the slope of the graph of $f$ is a positive integer greater than $5$ , what is the value of $k$ ?

Answer: 11

Answer

The value of

k

11

By equating the linear equations representing

f

and

g

, we find that the intersection point's

x

-coordinate

h

is given by

h = 6 + \frac{8m - 6}{m^2 + 1}

. Since

h

is an integer and

m

is a positive integer greater than

5

, the inequality

m^2 + 1 \le 8m - 6

limits

m

to the range

1 \le m \le 7

. Testing the integers greater than

5

(which are

6

and

7

) reveals that only

m = 7

yields an integer value for

h

, resulting in

h = 7

. Substituting these values back into the equation of either line yields

k = 11

Step-by-Step Solution

Write the equation of the linear function

f

in terms of its slope

m

y - 4 = m(x - 6) \implies y = mx + 4 - 6m

Since the graph of

f

passes through the point

(6, 4)

with slope

m

, we can use the point-slope form of a linear equation.

Write the equation of line

g

in slope-intercept form.

y = -\frac{1}{m}x + 12

Line

g

is perpendicular to the graph of

f

, so its slope is the negative reciprocal of

m

, which is

-\frac{1}{m}

. Its

y

-intercept is

(0, 12)

Equate the two equations to find the

x

-coordinate

h

of the intersection point

P(h, k)

in terms of

m

h = \frac{6m^2 + 8m}{m^2 + 1}

At the intersection point, the

y

-values of both equations are equal, so we solve

mx + 4 - 6m = -\frac{1}{m}x + 12

for

x

Analyze the expression for

h

to find the integer constraints on

m

h = 6 + \frac{8m - 6}{m^2 + 1} \implies m^2 - 8m + 7 \le 0 \implies 1 \le m \le 7

For

h

to be an integer,

\frac{8m - 6}{m^2 + 1}

must be an integer. Since

m

is a positive integer greater than

5

, the numerator is positive, meaning the denominator

m^2 + 1

must be less than or equal to the numerator

8m - 6

Test the possible integer values of

m

greater than

5

to find the one that yields integer coordinates.

m = 7 \implies h = 7

The only integers greater than

5

in the range

[1, 7]

are

6

and

7

. Testing

m=6

gives a non-integer, while

m=7

gives

h = 7

, which is an integer.

Calculate the

y

-coordinate

k

of the intersection point

P

k = 11

Substitute

h = 7

into the equation of line

g

k = -\frac{1}{7}(7) + 12 = 11

Key Concept

Linear Functions and Graphs

Question

Answer

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