To determine the correct system of equations, let's analyze the information given from the description you've provided:
- Solid line that passes through the point (2, 7).
- Dotted line intersects the first line at the point (-1, 1).
To find the equations of the lines, we can use the point-slope form of a linear equation, \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Given that we have two points for the solid line, (2, 7) and presumably another point that we can derive from the intersection (intersecting with the dotted line at (-1, 1)), we need to find the slopes.
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For the solid line that passes through (2, 7):
- We can determine the slope with another point (if there were any given or assumed).
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For the dotted line intersecting at (-1, 1):
- Its slope can also be determined if we had another point.
Now we can work through the options provided to see which one fits these conditions.
Let's Analyze the Options:
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Option 1: \[ \begin{align*} y &= 2x - 3 \ y &= 3x - 4 \end{align*} \]
- Check (2, 7) in the first equation: \(7 = 2(2) - 3\) => \(7 = 4 - 3\) => no, does not work.
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Option 2: \[ \begin{align*} y &= -x + 1 \ y &= 2x + 7 \end{align*} \]
- Check (2, 7) in second equation: \(7 = 2(2) + 7\) => \(7 = 4 + 7\) => no, does not work.
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Option 3: \[ \begin{align*} y &= 2x + 3 \ y &= 3x + 4 \end{align*} \]
- Check (2, 7) in the first equation: \(7 = 2(2) + 3\) => \(7 = 4 + 3\) => yes, works!
- Now for the other point (-1, 1): Check in second equation: \(1 = 3(-1) + 4\) => \(1 = -3 + 4\) => yes, works!
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Option 4: \[ \begin{align*} y &= 3x + 2 \ y &= 4x + 3 \end{align*} \]
- Check (2, 7) in the first equation: \(7 = 3(2) + 2\) => \(7 = 6 + 2\) => no, does not work.
Conclusion:
From the analysis, Option 3 is the correct system of equations.
\({ y = 2x + 3, y = 3x + 4 }\)