To analyze the given system of equations, we need to rewrite them in a standard form to determine their slopes and y-intercepts.
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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.
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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:
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The system has one solution. (False) Since the slopes are the same, the lines are parallel and do not intersect.
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The system consists of parallel lines. (True) Both lines have the same slope (\(\frac{1}{3}\)), making them parallel.
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Both lines have the same slope. (True) As calculated, both lines have a slope of \(\frac{1}{3}\).
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Both lines have the same y–intercept. (False) The y-intercepts are different: -4 and -\(\frac{7}{3}\).
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The equations represent the same line. (False) Because the y-intercepts are different, the lines are distinct.
True statements:
- The system consists of parallel lines.
- Both lines have the same slope.