In the standard (x, y) coordinate plane, line L has a y-intercept of 4. When line L is tra… — ACT

In the standard $(x, y)$ coordinate plane, line $L$ has a $y$ -intercept of $4$ . When line $L$ is translated horizontally to the right by $3$ units, the $y$ -intercept of the resulting line is $6$ units less than the $y$ -intercept of line $L$ . What is the $x$ -intercept of line $L$ ?

A
$2$
B
$-8$
$-2$ Answer
D
$-\frac{1}{2}$
E
$-6$

Answer

The

x

-intercept of line

L

-2

The original line

L

can be written as

y = mx + 4

. Translating the line horizontally to the right by

3

units replaces

x

with

x - 3

, resulting in the equation

y = m(x - 3) + 4 = mx - 3m + 4

. The

y

-intercept of this new line is

4 - 3m

. We are given that this new

y

-intercept is

6

units less than the original

y

-intercept of

4

, which gives the equation

4 - 3m = 4 - 6

. Solving for

m

yields

-3m = -6 \implies m = 2

. With a slope of

2

and a

y

-intercept of

4

, the equation of the original line

L

y = 2x + 4

. To find the

x

-intercept, we set

y = 0

, which gives

2x + 4 = 0 \implies x = -2

Step-by-Step Solution

Write the general equation of line

L

in slope-intercept form.

y = mx + 4

, where

m

is the slope of line

L

The

y

-intercept of line

L

is given as

4

, which represents the constant term

b

y = mx + b

Apply the horizontal translation of

3

units to the right.

y = m(x - 3) + 4

, which simplifies to

y = mx - 3m + 4

Translating a function horizontally to the right by

h

units replaces

x

with

x - h

Determine the slope

m

using the change in

y

-intercept.

The new

y

-intercept is

4 - 3m

. Since this is

6

units less than the original

y

-intercept of

4

, we set up the equation

4 - 3m = 4 - 6 \implies -3m = -6 \implies m = 2

We relate the algebraic expression for the new

y

-intercept to the given numerical relationship.

Find the

x

-intercept of line

L

by setting

y = 0

2x + 4 = 0 \implies 2x = -4 \implies x = -2

The

x

-intercept is the value of

x

when the line crosses the

x

-axis, meaning

y = 0

Key Concept

Linear Equations and Graphing

Question

Answer

Step-by-Step Solution

Key Concept

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