On a map of a town laid out on a standard coordinate grid, a straight path has an x-interc… — ACT

On a map of a town laid out on a standard coordinate grid, a straight path has an $x$ -intercept of $-6$ and a $y$ -intercept of $4$ . A landmark is located on this path at a point where the $x$ -coordinate is $9$ . What is the $y$ -coordinate of the landmark?

Answer: 10

Answer

The

y

-coordinate of the landmark is 10.

The correct answer is 10. The linear path passes through the points

(-6, 0)

and

(0, 4)

. Using the slope formula, the slope is

m = \frac{4 - 0}{0 - (-6)} = \frac{2}{3}

. The slope-intercept equation of the line is

y = \frac{2}{3}x + 4

. Substituting the landmark's

x

-coordinate of

9

into the equation gives

y = \frac{2}{3}(9) + 4 = 10

Step-by-Step Solution

Identify the coordinates of the points representing the intercepts on the coordinate grid.

The points are

(-6, 0)

and

(0, 4)

x

-intercept of

-6

corresponds to the coordinate point

(-6, 0)

, and a

y

-intercept of

4

corresponds to the coordinate point

(0, 4)

Calculate the slope of the linear path.

The slope is

\frac{2}{3}

Using the slope formula

m = \frac{y_2 - y_1}{x_2 - x_1}

, substitute the points

(-6, 0)

and

(0, 4)

to get

m = \frac{4 - 0}{0 - (-6)} = \frac{4}{6} = \frac{2}{3}

Write the linear equation representing the path.

The equation is

y = \frac{2}{3}x + 4

With a slope of

m = \frac{2}{3}

and a

y

-intercept of

b = 4

, the equation in slope-intercept form (

y = mx + b

) is

y = \frac{2}{3}x + 4

Find the

y

-coordinate of the landmark by substituting its

x

-coordinate.

The

y

-coordinate is 10.

Substitute

x = 9

into the equation:

y = \frac{2}{3}(9) + 4 = 6 + 4 = 10

Key Concept

Linear Equations and Graphing

Question

Answer

Step-by-Step Solution

Key Concept

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