Question
Use the image to answer the question.
Two lines slope upward on a coordinate plane. A solid line passes through the plotted point left parenthesis 2 comma 7 right parenthesis. A dotted line intersects the first line at the plotted point left parenthesis negative 1 comma 1 right parenthesis.
What is the system of equations shown in the graph?
(1 point)
Responses
{y=2x−3y=3x−4
enlraged left brace start layout first row y equals 2 x minus 3 second row y equals 3 x minus 4 end layout
{y=−x+1y=2x+7
enlarged left brace start layout first row y equals negative x plus 1 second row y equals 2 x plus 7 end layout
{y=2x+3y=3x+4
enlarged left brace start layout first row y equals 2 x plus 3 second row y equals 3 x plus 4 end layout
{y=3x+2y=4x+3
Two lines slope upward on a coordinate plane. A solid line passes through the plotted point left parenthesis 2 comma 7 right parenthesis. A dotted line intersects the first line at the plotted point left parenthesis negative 1 comma 1 right parenthesis.
What is the system of equations shown in the graph?
(1 point)
Responses
{y=2x−3y=3x−4
enlraged left brace start layout first row y equals 2 x minus 3 second row y equals 3 x minus 4 end layout
{y=−x+1y=2x+7
enlarged left brace start layout first row y equals negative x plus 1 second row y equals 2 x plus 7 end layout
{y=2x+3y=3x+4
enlarged left brace start layout first row y equals 2 x plus 3 second row y equals 3 x plus 4 end layout
{y=3x+2y=4x+3
Answers
GPT-4o mini
To determine the correct system of equations, let's analyze the information given from the description you've provided:
1. **Solid line that passes through the point (2, 7)**.
2. **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.
1. **For the solid line** that passes through (2, 7):
- We can determine the slope with another point (if there were any given or assumed).
2. **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:
1. **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.
2. **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.
3. **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!
4. **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.
\(\{ y = 2x + 3, y = 3x + 4 \}\)
1. **Solid line that passes through the point (2, 7)**.
2. **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.
1. **For the solid line** that passes through (2, 7):
- We can determine the slope with another point (if there were any given or assumed).
2. **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:
1. **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.
2. **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.
3. **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!
4. **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.
\(\{ y = 2x + 3, y = 3x + 4 \}\)