Question
Which graph shows the solution to the system of linear equations?
y = 3x − 3
3x − 3y = −3
Coordinate plane with one line that passes through the points 0 comma negative 3 and 1 comma 0.
Coordinate plane with one line that passes through the points 0 comma 1 and 1 comma 2.
Coordinate plane with one line that passes through the points 0 comma 1 and 1 comma 2 and another line that passes through the points 0 comma negative 3 and 1 comma 0.
Coordinate plane with one line that passes through the points 0 comma negative 1 and 1 comma 2 and another line that passes through the points 0 comma negative 3 and 1 comma 0.
y = 3x − 3
3x − 3y = −3
Coordinate plane with one line that passes through the points 0 comma negative 3 and 1 comma 0.
Coordinate plane with one line that passes through the points 0 comma 1 and 1 comma 2.
Coordinate plane with one line that passes through the points 0 comma 1 and 1 comma 2 and another line that passes through the points 0 comma negative 3 and 1 comma 0.
Coordinate plane with one line that passes through the points 0 comma negative 1 and 1 comma 2 and another line that passes through the points 0 comma negative 3 and 1 comma 0.
Answers
GPT-4o mini
To solve the system of equations given by \(y = 3x - 3\) and \(3x - 3y = -3\), we can first analyze each equation.
1. **First Equation:** \(y = 3x - 3\) is already in slope-intercept form \(y = mx + b\), where \(m = 3\) and \(b = -3\). This tells us the line has a slope of 3 and a y-intercept at (0, -3).
2. **Second Equation:** We need to rearrange \(3x - 3y = -3\) into slope-intercept form.
- Start with the original equation:
\[
3x - 3y = -3
\]
- Isolate \(y\):
\[
-3y = -3x - 3
\]
- Divide by -3 to solve for \(y\):
\[
y = x + 1
\]
This means the second line has a slope of 1 and a y-intercept at (0, 1).
Now we have two lines to work with:
1. \(y = 3x - 3\) (line 1)
- Passes through points (0, -3) and (1, 0).
2. \(y = x + 1\) (line 2)
- Passes through points (0, 1) and (1, 2).
By analyzing the answer options:
- The first option shows a single line passing through (0, -3) and (1, 0)—this corresponds to the first equation only.
- The second option also shows a single line passing through (0, 1) and (1, 2)—this corresponds to the second equation only.
- The third option shows two lines: one passing through (0, 1) and (1, 2) and the other through (0, -3) and (1, 0). This correctly represents both equations.
- The fourth option shows two lines, but one does not correlate to the equations given.
Thus, **the correct answer is the third option:** "Coordinate plane with one line that passes through the points (0, 1) and (1, 2) and another line that passes through the points (0, -3) and (1, 0)."
1. **First Equation:** \(y = 3x - 3\) is already in slope-intercept form \(y = mx + b\), where \(m = 3\) and \(b = -3\). This tells us the line has a slope of 3 and a y-intercept at (0, -3).
2. **Second Equation:** We need to rearrange \(3x - 3y = -3\) into slope-intercept form.
- Start with the original equation:
\[
3x - 3y = -3
\]
- Isolate \(y\):
\[
-3y = -3x - 3
\]
- Divide by -3 to solve for \(y\):
\[
y = x + 1
\]
This means the second line has a slope of 1 and a y-intercept at (0, 1).
Now we have two lines to work with:
1. \(y = 3x - 3\) (line 1)
- Passes through points (0, -3) and (1, 0).
2. \(y = x + 1\) (line 2)
- Passes through points (0, 1) and (1, 2).
By analyzing the answer options:
- The first option shows a single line passing through (0, -3) and (1, 0)—this corresponds to the first equation only.
- The second option also shows a single line passing through (0, 1) and (1, 2)—this corresponds to the second equation only.
- The third option shows two lines: one passing through (0, 1) and (1, 2) and the other through (0, -3) and (1, 0). This correctly represents both equations.
- The fourth option shows two lines, but one does not correlate to the equations given.
Thus, **the correct answer is the third option:** "Coordinate plane with one line that passes through the points (0, 1) and (1, 2) and another line that passes through the points (0, -3) and (1, 0)."