In the xy-plane, a line passes through the origin and has a slope of \frac{3}{5}. If the p… — SAT

In the $xy$ -plane, a line passes through the origin and has a slope of $\frac{3}{5}$ . If the point $(k, 9)$ lies on the line, what is the value of $k$ ?

Answer: 15

Answer

A line passing through the origin has a

y

-intercept of

0

. Thus, its equation in slope-intercept form is

y = mx

, where

m

is the slope. Given the slope is

\frac{3}{5}

, the equation is

y = \frac{3}{5}x

. Since the point

(k, 9)

lies on the line, substituting these coordinates yields

9 = \frac{3}{5}k

. Multiplying both sides by

\frac{5}{3}

gives

k = 15

Step-by-Step Solution

Write the equation of the line in slope-intercept form.

y = \frac{3}{5}x

The line passes through the origin

(0,0)

, so the

y

-intercept is

0

, and the slope is

\frac{3}{5}

Substitute the coordinates of the point

(k, 9)

into the line's equation.

9 = \frac{3}{5}k

A point lies on a line if its coordinates satisfy the equation of the line.

Solve the equation for

k

k = 15

Multiply both sides of the equation by

\frac{5}{3}

to isolate

k

Key Concept

Using the slope and a point on a line to determine an unknown coordinate.

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