Question

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

y = –Three-fourthsx + 1
3x − 4y = −4
4x − 3y = −3
y – 2 = –Three-fourths(x – 4)
y + 2 = Three-fourths(x + 4)

Answer 1

To find the equations that represent the line parallel to \(3x - 4y = 7\) and passing through the point \((-4, -2)\), we first need to determine the slope of the given line.

1. **Find the slope of the given line**:
We can rewrite the equation \(3x - 4y = 7\) in slope-intercept form \(y = mx + b\), where \(m\) is the slope.
\[
-4y = -3x + 7
\]
\[
y = \frac{3}{4}x - \frac{7}{4}
\]
The slope is \(\frac{3}{4}\).

2. **Find the slope of the parallel line**:
Since parallel lines have the same slope, the slope of our new line that will pass through \((-4, -2)\) is also \(\frac{3}{4}\).

3. **Use point-slope form to find the equation of the parallel line**:
The point-slope form of a line is given by:
\[
y - y_1 = m(x - x_1)
\]
Substituting in the point \((-4, -2)\) and the slope \(\frac{3}{4}\):
\[
y - (-2) = \frac{3}{4}(x - (-4))
\]
\[
y + 2 = \frac{3}{4}(x + 4)
\]
This forms one of the equations.

4. **Check the candidates**:
- \(y + 2 = \frac{3}{4}(x + 4)\) (This is the line we just derived).
- \(y = -\frac{3}{4}x + 1\) (This has a slope of \(-\frac{3}{4}\), not valid).
- \(3x - 4y = -4\) (We will check if it's parallel).
- \(4x - 3y = -3\) (We will check if it's parallel).
- \(y - 2 = -\frac{3}{4}(x - 4)\) (This has a slope of \(-\frac{3}{4}\), not valid).

5. **Check equations for parallelism**:
- For \(3x - 4y = -4\):
\[
-4y = -3x - 4 \implies y = \frac{3}{4}x + 1
\]
(This line is parallel because it has the same slope \(\frac{3}{4}\)).

- For \(4x - 3y = -3\):
\[
-3y = -4x - 3 \implies y = \frac{4}{3}x + 1
\]
(This line is not parallel because its slope is \(\frac{4}{3}\)).

Therefore, the two equations that represent the line that is parallel to \(3x − 4y = 7\) and passes through the point \((-4, -2)\) are:

- \(y + 2 = \frac{3}{4}(x + 4)\)
- \(3x - 4y = -4\)

These are the correct options.