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 given system of equations, we need to rewrite them in a standard form to determine their slopes and y-intercepts.

1. The first equation is:

\[
y = \frac{1}{3}x - 4
\]

This is in slope-intercept form \(y = mx + b\) where \(m\) is the slope and \(b\) is the y-intercept. Here, the slope (m) is \(\frac{1}{3}\) and the y-intercept (b) is -4.

2. The second equation is:

\[
3y - x = -7
\]

We can rearrange this equation into slope-intercept form.

Start by adding \(x\) to both sides:

\[
3y = x - 7
\]

Then divide everything by 3:

\[
y = \frac{1}{3}x - \frac{7}{3}
\]

Here, the slope is also \(\frac{1}{3}\) and the y-intercept is \(-\frac{7}{3}\).

**Now we can analyze the statements:**

- **The system has one solution.** (False) Since the slopes are the same, the lines are parallel and do not intersect.

- **The system consists of parallel lines.** (True) Both lines have the same slope (\(\frac{1}{3}\)), making them parallel.

- **Both lines have the same slope.** (True) As calculated, both lines have a slope of \(\frac{1}{3}\).

- **Both lines have the same y–intercept.** (False) The y-intercepts are different: -4 and -\(\frac{7}{3}\).

- **The equations represent the same line.** (False) Because the y-intercepts are different, the lines are distinct.

**True statements:**
1. The system consists of parallel lines.
2. Both lines have the same slope.