Question
Graph the system of equations to determine the solution. In your final answer, include your graph and the solution.
y=x-1
-x+3/2=-y
y=x-1
-x+3/2=-y
Answers
Answered by
GPT-4o mini
To graph the system of equations, let's first rewrite both equations in slope-intercept form (y = mx + b) for ease of graphing.
1. The first equation is already in slope-intercept form:
\[
y = x - 1
\]
2. The second equation is \(-x + \frac{3}{2} = -y\). We can rearrange it to solve for \(y\):
\[
-y = -x + \frac{3}{2}
\]
\[
y = x - \frac{3}{2}
\]
Next, let's summarize the equations:
- Equation 1: \(y = x - 1\)
- Equation 2: \(y = x - \frac{3}{2}\)
### Graphing the Equations
1. **Graph \(y = x - 1\)**:
- This line has a slope of 1 and a y-intercept of -1. It passes through points like (0, -1) and (1, 0).
2. **Graph \(y = x - \frac{3}{2}\)**:
- This line also has a slope of 1 but a y-intercept of -1.5. It passes through points like (0, -1.5) and (1, -0.5).
### Finding the Intersection
To find the solution to the system, we can set the two equations equal to each other:
\[
x - 1 = x - \frac{3}{2}
\]
Subtract \(x\) from both sides:
\[
-1 = -\frac{3}{2}
\]
This equation results in a contradiction, meaning there are no \(x\) values that satisfy both equations. Analyzing the slopes, we see both lines have a slope of 1, meaning they are parallel and thus will never intersect.
### Conclusion
Since the two lines are parallel, there is no solution to this system of equations.
### Graph Representation
Unfortunately, I can't create visual graphics directly. However, here's a description of how to graph it:
1. **Draw two lines**:
- For \(y = x - 1\), plot points (0, -1) and (1, 0) and draw the line.
- For \(y = x - \frac{3}{2}\), plot points (0, -1.5) and (1, -0.5) and draw the line.
2. **Label the lines**:
- Label the first line as \(y = x - 1\) and the second as \(y = x - \frac{3}{2}\).
Since the lines are parallel, you would visibly observe that they do not touch or intersect anywhere on the graph.
The final answer is:
There is **no solution** to the system as the lines are parallel.
1. The first equation is already in slope-intercept form:
\[
y = x - 1
\]
2. The second equation is \(-x + \frac{3}{2} = -y\). We can rearrange it to solve for \(y\):
\[
-y = -x + \frac{3}{2}
\]
\[
y = x - \frac{3}{2}
\]
Next, let's summarize the equations:
- Equation 1: \(y = x - 1\)
- Equation 2: \(y = x - \frac{3}{2}\)
### Graphing the Equations
1. **Graph \(y = x - 1\)**:
- This line has a slope of 1 and a y-intercept of -1. It passes through points like (0, -1) and (1, 0).
2. **Graph \(y = x - \frac{3}{2}\)**:
- This line also has a slope of 1 but a y-intercept of -1.5. It passes through points like (0, -1.5) and (1, -0.5).
### Finding the Intersection
To find the solution to the system, we can set the two equations equal to each other:
\[
x - 1 = x - \frac{3}{2}
\]
Subtract \(x\) from both sides:
\[
-1 = -\frac{3}{2}
\]
This equation results in a contradiction, meaning there are no \(x\) values that satisfy both equations. Analyzing the slopes, we see both lines have a slope of 1, meaning they are parallel and thus will never intersect.
### Conclusion
Since the two lines are parallel, there is no solution to this system of equations.
### Graph Representation
Unfortunately, I can't create visual graphics directly. However, here's a description of how to graph it:
1. **Draw two lines**:
- For \(y = x - 1\), plot points (0, -1) and (1, 0) and draw the line.
- For \(y = x - \frac{3}{2}\), plot points (0, -1.5) and (1, -0.5) and draw the line.
2. **Label the lines**:
- Label the first line as \(y = x - 1\) and the second as \(y = x - \frac{3}{2}\).
Since the lines are parallel, you would visibly observe that they do not touch or intersect anywhere on the graph.
The final answer is:
There is **no solution** to the system as the lines are parallel.