A line graphed on the (x, y) coordinate plane has a y-intercept of -1 and passes through t… — ACT

A line graphed on the $(x, y)$ coordinate plane has a $y$ -intercept of $-1$ and passes through the point $(3, 8)$ . If the point $(a, 14)$ also lies on this line, what is the value of $a$ ?

Answer: 5

Answer

The line has a y-intercept of

-1

, which corresponds to the point

(0, -1)

. The slope is calculated as

m = \frac{8 - (-1)}{3 - 0} = 3

. The equation of the line is

y = 3x - 1

. Substituting

(a, 14)

into this equation gives

14 = 3a - 1

, which simplifies to

15 = 3a

, and solving for

a

yields

5

Step-by-Step Solution

Identify the coordinate representation of the y-intercept.

The point is

(0, -1)

A y-intercept of

-1

means the line crosses the y-axis at the point where

x = 0

, which is

(0, -1)

Calculate the slope of the line.

m = 3

Using the slope formula

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

with the points

(0, -1)

and

(3, 8)

, we get

m = \frac{8 - (-1)}{3 - 0} = 3

Write the equation of the line and solve for

a

a = 5

The equation in slope-intercept form is

y = 3x - 1

. Substituting the point

(a, 14)

yields

14 = 3a - 1

. Adding 1 to both sides gives

15 = 3a

, so

a = 5

Key Concept

Finding the equation of a line from two points and solving for a missing coordinate.

Estimated Time:1m 30s

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