Question

Which statements about the system are true? Select two options.

y = y equals StartFraction 1 over 3 EndFraction x minus 4.x – 4

3y – x = –7

The system has one solution.
The system consists of parallel lines.
Both lines have the same slope.
Both lines have the same y–intercept.
The equations represent the same line.

Answer 1

To analyze the provided system of equations, let's first rewrite both equations in a more standard form.

1. The first equation is given as:
\[
y = \frac{1}{3}x - 4
\]
This has a slope (\(m\)) of \(\frac{1}{3}\) and a \(y\)-intercept of \(-4\).

2. The second equation is:
\[
3y - x = -7
\]
We can rearrange it into slope-intercept form (\(y = mx + b\)):
- First, isolate \(3y\):
\[
3y = x - 7
\]
- Now, divide by 3:
\[
y = \frac{1}{3}x - \frac{7}{3}
\]
This also has a slope of \(\frac{1}{3}\) but a different \(y\)-intercept of \(-\frac{7}{3}\).

Now let's evaluate the statements:

1. **The system has one solution.**
- **False.** Since the lines have the same slope but different \(y\)-intercepts, they are parallel lines and will never intersect.

2. **The system consists of parallel lines.**
- **True.** Both lines have the same slope (\(\frac{1}{3}\)) but different \(y\)-intercepts, confirming that they are parallel.

3. **Both lines have the same slope.**
- **True.** Both equations have the slope of \(\frac{1}{3}\).

4. **Both lines have the same y–intercept.**
- **False.** The first line has a \(y\)-intercept of \(-4\) and the second has a \(y\)-intercept of \(-\frac{7}{3}\), which are not equal.

5. **The equations represent the same line.**
- **False.** They are different lines (as established by different \(y\)-intercepts).

Thus, the two statements that are true are:

- **The system consists of parallel lines.**
- **Both lines have the same slope.**