To analyze the provided system of equations, let's first rewrite both equations in a more standard form.
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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 \(-4\).
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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:
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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.
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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.
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Both lines have the same slope.
- True. Both equations have the slope of \(\frac{1}{3}\).
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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.
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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.