In the standard (x, y) coordinate plane, the line described by the equation 2x - 3y = 10 i… — ACT

In the standard $(x, y)$ coordinate plane, the line described by the equation $2x - 3y = 10$ is translated $2$ units to the left and $1$ unit up. What is the $y$ -intercept of the translated line?

A
$-\frac{17}{3}$
B
$-3$
C
$-\frac{7}{3}$
D
$-\frac{5}{3}$
$-1$ Answer

Answer

The correct answer is

-1

The correct answer is

-1

. Translating a line 2 units to the left shifts every point on the line by

-2

in the

x

-direction, which is represented algebraically by substituting

x + 2

for

x

. Translating it 1 unit up shifts every point by

+1

in the

y

-direction, represented by substituting

y - 1

for

y

. Substituting these into the original equation

2x - 3y = 10

yields

2(x + 2) - 3(y - 1) = 10

. Simplifying this equation gives

2x + 4 - 3y + 3 = 10

, which becomes

2x - 3y = 3

. To find the

y

-intercept of this new line, we set

x = 0

, giving

-3y = 3

, which solves to

y = -1

. Alternatively, we can find the original

y

-intercept of

(0, -10/3)

. Under the translation, this point shifts to

(0 - 2, -10/3 + 1) = (-2, -7/3)

. Since the slope of the line remains

2/3

, the new line equation is

y - (-7/3) = (2/3)(x - (-2))

, which simplifies to

y = (2/3)x - 1

, confirming the

y

-intercept is

-1

Step-by-Step Solution

Express the translation of the coordinates mathematically. Translating a line 2 units to the left replaces

x

with

x + 2

. Translating it 1 unit up replaces

y

with

y - 1

The substitution expressions are

x \to x + 2

and

y \to y - 1

This sets up the algebraic transformation of the original line's equation.

Substitute the expressions into the original equation

2x - 3y = 10

and simplify.

2(x + 2) - 3(y - 1) = 10 \implies 2x + 4 - 3y + 3 = 10 \implies 2x - 3y + 7 = 10 \implies 2x - 3y = 3

This derives the equation of the translated line.

Find the

y

-intercept of the translated line

2x - 3y = 3

by setting

x = 0

2(0) - 3y = 3 \implies -3y = 3 \implies y = -1

The

y

-intercept is the point on the line where

x = 0

Key Concept

Translating linear equations in the standard coordinate plane

Estimated Time:1m 30s

Question

Answer

Step-by-Step Solution

Key Concept

Common Mistakes

Topics

More questions from this topic