Which equations represent the line that is parallel to [tex]3x - 4y = 7[/tex] and passes through the point [tex](-4, -2)[/tex]? Select two options.

A. [tex]y = -\frac{3}{4} x + 1[/tex]

B. [tex]3x - 4y = -4[/tex]

C. [tex]4x - 3y = -3[/tex]

D. [tex]y - 2 = -\frac{3}{4}(x - 4)[/tex]

E. [tex]y + 2 = \frac{3}{4}(x + 4)[/tex]



Answer :

Let's start by determining the slope of the given line [tex]\(3x - 4y = 7\)[/tex]. To find the slope, we'll rewrite this equation in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope.

First, solve for [tex]\(y\)[/tex]:

[tex]\[ 3x - 4y = 7 \][/tex]

Subtract [tex]\(3x\)[/tex] from both sides:

[tex]\[ -4y = -3x + 7 \][/tex]

Divide every term by [tex]\(-4\)[/tex]:

[tex]\[ y = \frac{3}{4}x - \frac{7}{4} \][/tex]

The slope [tex]\(m\)[/tex] of the given line is [tex]\(\frac{3}{4}\)[/tex].

Any line parallel to this one will have the same slope, [tex]\(\frac{3}{4}\)[/tex]. Now, let's examine each of the given options to see if they both have this slope and go through the point [tex]\((-4, -2)\)[/tex]:

1. Option 1: [tex]\(y = -\frac{3}{4}x + 1\)[/tex]

The slope of this line is [tex]\(-\frac{3}{4}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.

2. Option 2: [tex]\(3x - 4y = -4\)[/tex]

To find the slope, we can rearrange this equation into slope-intercept form:

[tex]\[ 3x - 4y = -4 \][/tex]

Subtract [tex]\(3x\)[/tex] from both sides:

[tex]\[ -4y = -3x - 4 \][/tex]

Divide every term by [tex]\(-4\)[/tex]:

[tex]\[ y = \frac{3}{4}x + 1 \][/tex]

The slope is [tex]\(\frac{3}{4}\)[/tex], which matches the slope of the given line. Now we need to check if it passes through the point [tex]\((-4, -2)\)[/tex]. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -2\)[/tex] into the equation [tex]\(y = \frac{3}{4}x + 1\)[/tex]:

[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]

Simplify:

[tex]\[ -2 = -3 + 1 \][/tex]

[tex]\[ -2 = -2 \][/tex]

This is true, so Option 2 is correct.

3. Option 3: [tex]\(4x - 3y = -3\)[/tex]

To find the slope, we can rearrange this equation into slope-intercept form:

[tex]\[ 4x - 3y = -3 \][/tex]

Subtract [tex]\(4x\)[/tex] from both sides:

[tex]\[ -3y = -4x - 3 \][/tex]

Divide every term by [tex]\(-3\)[/tex]:

[tex]\[ y = \frac{4}{3}x + 1 \][/tex]

The slope is [tex]\(\frac{4}{3}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.

4. Option 4: [tex]\(y - 2 = -\frac{3}{4}(x - 4)\)[/tex]

First, convert this into slope-intercept form:

Distribute [tex]\(-\frac{3}{4}\)[/tex]:

[tex]\[ y - 2 = -\frac{3}{4}x + 3 \][/tex]

Add 2 to both sides:

[tex]\[ y = -\frac{3}{4}x + 5 \][/tex]

The slope is [tex]\(-\frac{3}{4}\)[/tex], which is different from [tex]\(\frac{3}{4}\)[/tex]. Thus, this line is not parallel to the given line.

5. Option 5: [tex]\(y + 2 = \frac{3}{4}(x + 4)\)[/tex]

First, convert this into slope-intercept form:

Distribute [tex]\(\frac{3}{4}\)[/tex]:

[tex]\[ y + 2 = \frac{3}{4}x + 3 \][/tex]

Subtract 2 from both sides:

[tex]\[ y = \frac{3}{4}x + 1 \][/tex]

The slope is [tex]\(\frac{3}{4}\)[/tex], which matches the slope of the given line. Now we need to check if it passes through the point [tex]\((-4, -2)\)[/tex]. Substitute [tex]\(x = -4\)[/tex] and [tex]\(y = -2\)[/tex] into the equation [tex]\(y = \frac{3}{4}x + 1\)[/tex]:

[tex]\[ -2 = \frac{3}{4}(-4) + 1 \][/tex]

Simplify:

[tex]\[ -2 = -3 + 1 \][/tex]

[tex]\[ -2 = -2 \][/tex]

This is true, so Option 5 is correct.

Thus, the equations that represent the line parallel to [tex]\(3x - 4y = 7\)[/tex] and pass through the point [tex]\((-4, -2)\)[/tex] are Options 2 and 5.