Question
Solve the system of equations by graphing.
How many solutions?
y =-2x + 3
-2x = y +1
How many solutions?
y =-2x + 3
-2x = y +1
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To solve the system of equations by graphing, we'll first rewrite both equations in slope-intercept form (y = mx + b) and then plot them.
1. The first equation is already in slope-intercept form:
\[
y = -2x + 3
\]
This has a slope (m) of -2 and a y-intercept (b) of 3.
2. For the second equation, we'll rearrange it to solve for y:
\[
-2x = y + 1 \\
y = -2x - 1
\]
This equation also has a slope of -2 but a different y-intercept of -1.
Now we can graph both equations:
- For \(y = -2x + 3\):
- If \(x = 0\), then \(y = 3\) (point: (0, 3))
- If \(x = 1\), then \(y = 1\) (point: (1, 1))
- For \(y = -2x - 1\):
- If \(x = 0\), then \(y = -1\) (point: (0, -1))
- If \(x = 1\), then \(y = -3\) (point: (1, -3))
When we plot these points, we see that both lines have the same slope (-2), which means they are parallel. Since parallel lines never intersect, there are no solutions to this system of equations.
**Conclusion**: The system of equations has **no solutions**.
1. The first equation is already in slope-intercept form:
\[
y = -2x + 3
\]
This has a slope (m) of -2 and a y-intercept (b) of 3.
2. For the second equation, we'll rearrange it to solve for y:
\[
-2x = y + 1 \\
y = -2x - 1
\]
This equation also has a slope of -2 but a different y-intercept of -1.
Now we can graph both equations:
- For \(y = -2x + 3\):
- If \(x = 0\), then \(y = 3\) (point: (0, 3))
- If \(x = 1\), then \(y = 1\) (point: (1, 1))
- For \(y = -2x - 1\):
- If \(x = 0\), then \(y = -1\) (point: (0, -1))
- If \(x = 1\), then \(y = -3\) (point: (1, -3))
When we plot these points, we see that both lines have the same slope (-2), which means they are parallel. Since parallel lines never intersect, there are no solutions to this system of equations.
**Conclusion**: The system of equations has **no solutions**.
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