Solution Sets of Simultaneous Equations Quick Check

Question

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Question
Which graph shows the solution of −3x−y=3 and y=−3x+4?(1 point)
Responses

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.

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.
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Answer 1

To determine which graph shows the solution to the equations \(-3x - y = 3\) and \(y = -3x + 4\), we can analyze the two equations to see if they intersect, are parallel, or are identical.

1. Convert both equations to slope-intercept form (y = mx + b):

   • The first equation is \(-3x - y = 3\). We can rearrange it to find y: \[ -y = 3 + 3x \implies y = -3x - 3 \]

   • The second equation is already in slope-intercept form: \(y = -3x + 4\).

2. Identify the slopes and y-intercepts:

   • From \(y = -3x - 3\): the slope (m) is \(-3\) and the y-intercept (b) is \(-3\).
   • From \(y = -3x + 4\): the slope is \(-3\) and the y-intercept is \(4\).

3. Analyze the lines:

   • Both lines have the same slope of \(-3\), which means they are parallel.
   • They have different y-intercepts, so they will never intersect.

Given that the two lines are parallel, the correct response is "Two parallel lines are graphed across a coordinate plane. One line is dotted while the other is solid."