A line has a slope of -3. If the sum of its x-intercept and its y-intercept is 12, what is… — ACT

A line has a slope of $-3$ . If the sum of its $x$ -intercept and its $y$ -intercept is $12$ , what is the $y$ -intercept of the line?

Answer: 9

Answer

The y-intercept of the line is 9.

The equation of the line is

y = -3x + b

, where

b

is the

y

-intercept. The

x

-intercept is found by setting

y=0

, which yields

x = \frac{b}{3}

. Setting their sum to

12

gives

\frac{b}{3} + b = 12

, which simplifies to

4b = 36

and results in

b = 9

Step-by-Step Solution

Express the equation of the line using the slope-intercept form.

The equation of the line is

y = -3x + b

, where

b

is the

y

-intercept.

We are given that the slope of the line is

-3

Find the

x

-intercept of the line in terms of

b

Setting

y = 0

gives

0 = -3x + b \implies 3x = b \implies x = \frac{b}{3}

. Thus, the

x

-intercept is

\frac{b}{3}

The

x

-intercept of a line is the

x

-coordinate where the line crosses the

x

-axis, which occurs when

y = 0

Set up an equation using the given sum of the intercepts.

\frac{b}{3} + b = 12

The problem states that the sum of the

x

-intercept and the

y

-intercept is

12

Solve the linear equation for the

y

-intercept

b

Multiply the entire equation by

3

to clear the denominator:

b + 3b = 36 \implies 4b = 36 \implies b = 9

Solving this equation yields the value of the

y

-intercept.

Key Concept

Linear equations, slope-intercept form, and finding intercepts.

Alternative Method

Alternatively, you can write the equation of the line in intercept form:

\frac{x}{a} + \frac{y}{b} = 1

, where

a

is the

x

-intercept and

b

is the

y

-intercept. The slope of this line is

-\frac{b}{a} = -3

, which means

b = 3a

. Since the sum of the intercepts is

12

, we write

a + b = 12

. Substituting

b = 3a

into this sum gives

a + 3a = 12 \implies 4a = 12 \implies a = 3

. Therefore, the

y

-intercept

b

3(3) = 9

Estimated Time:1m 30s

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Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Alternative Method

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