Question
Which graph shows the solution of −3x−y=3 and y=−3x+4?(1 point)
Responses
Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
Image with alt text: Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
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Responses
Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
Image with alt text: Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
Two intersecting lines are graphed across a coordinate plane.
Image with alt text: Two intersecting lines are graphed across a coordinate plane.
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Answers
GPT-4o mini
To determine which graph represents the solution to the equations \(-3x - y = 3\) and \(y = -3x + 4\), we need to analyze both equations.
1. **First equation (\(-3x - y = 3\))**:
- We can rearrange it into slope-intercept form (\(y = mx + b\)):
\[
-y = 3 + 3x
\]
\[
y = -3x - 3
\]
So, the first line has a slope of -3 and a y-intercept of -3.
2. **Second equation (\(y = -3x + 4\))**:
- This is already in slope-intercept form, where the slope is -3 and the y-intercept is 4.
Now we observe that both lines have the same slope (-3), which means they are parallel. The difference in y-intercepts (-3 and 4) confirms that they will never intersect.
Based on this analysis, the correct graph must show two parallel lines. Therefore, the response that indicates **"Two parallel lines are graphed across a coordinate plane"** is the correct answer.
1. **First equation (\(-3x - y = 3\))**:
- We can rearrange it into slope-intercept form (\(y = mx + b\)):
\[
-y = 3 + 3x
\]
\[
y = -3x - 3
\]
So, the first line has a slope of -3 and a y-intercept of -3.
2. **Second equation (\(y = -3x + 4\))**:
- This is already in slope-intercept form, where the slope is -3 and the y-intercept is 4.
Now we observe that both lines have the same slope (-3), which means they are parallel. The difference in y-intercepts (-3 and 4) confirms that they will never intersect.
Based on this analysis, the correct graph must show two parallel lines. Therefore, the response that indicates **"Two parallel lines are graphed across a coordinate plane"** is the correct answer.