To analyze the system of equations given:
- Equation 1: \( 2y = x + 10 \)
- Equation 2: \( 3y = 3x + 15 \)
First, let's rearrange both equations into the slope-intercept form \( y = mx + b \), which makes it easier to compare them.
For the first equation:
\[ 2y = x + 10 \] Dividing by 2 gives: \[ y = \frac{1}{2}x + 5 \]
For the second equation:
\[ 3y = 3x + 15 \] Dividing by 3 gives: \[ y = x + 5 \]
Now we can analyze the slopes and y-intercepts of both equations:
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For the first equation \( y = \frac{1}{2}x + 5 \):
- Slope \( m_1 = \frac{1}{2} \)
- y-intercept \( b_1 = 5 \)
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For the second equation \( y = x + 5 \):
- Slope \( m_2 = 1 \)
- y-intercept \( b_2 = 5 \)
Now we can check the true statements:
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The system has one solution.
- False. The lines have different slopes, so they intersect at exactly one point.
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The system graphs parallel lines.
- False. The lines are not parallel, as they have different slopes.
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Both lines have the same slope.
- False. They have slopes of \( \frac{1}{2} \) and \( 1 \) respectively, which are different.
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Both lines have the same y-intercept.
- True. Both equations have a y-intercept of 5.
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The equations graph the same line.
- False. They have different slopes, hence they are not the same line.
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The solution is the intersection of the 2 lines.
- True. Since the lines intersect at one point, that point is the solution to the system.
Based on this analysis, the true statements are:
- Both lines have the same y-intercept.
- The solution is the intersection of the 2 lines.
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