Answer
The system of equations has one solution at the point (2, 0).
Solution
Certainly! Let's analyze and graph the system of equations you've provided.
## System of Equations
You have the following system:
\[
\begin{cases}
2x + y = 4 \\
y = \frac{1}{2}x - 1
\end{cases}
\]
## Solving the System
To find the solution to this system, we'll use the substitution method.
1. **From the second equation**, we already have \( y \) expressed in terms of \( x \):
\[
y = \frac{1}{2}x - 1
\]
2. **Substitute** this expression for \( y \) into the first equation:
\[
2x + \left(\frac{1}{2}x - 1\right) = 4
\]
3. **Combine like terms**:
\[
2x + \frac{1}{2}x - 1 = 4
\]
\[
\frac{5}{2}x - 1 = 4
\]
4. **Solve for \( x \)**:
\[
\frac{5}{2}x = 5
\]
\[
x = 2
\]
5. **Find \( y \)** using the second equation:
\[
y = \frac{1}{2}(2) - 1 = 1 - 1 = 0
\]
**Solution:** The system has a single solution at the point \( (2, 0) \).
## Graphing the System
Let's graph both equations to visualize their intersection.
### 1. Graph of \( 2x + y = 4 \)
Rearrange to slope-intercept form (\( y = mx + b \)):
\[
y = -2x + 4
\]
- **Slope (m):** -2
- **Y-intercept (b):** 4
**Points to Plot:**
- When \( x = 0 \), \( y = 4 \) → Point \( (0, 4) \)
- When \( x = 2 \), \( y = 0 \) → Point \( (2, 0) \)
### 2. Graph of \( y = \frac{1}{2}x - 1 \)
- **Slope (m):** \( \frac{1}{2} \)
- **Y-intercept (b):** -1
**Points to Plot:**
- When \( x = 0 \), \( y = -1 \) → Point \( (0, -1) \)
- When \( x = 2 \), \( y = 0 \) → Point \( (2, 0) \)
### Graph Illustration
Below is a graphical representation of the system:
```plaintext
y
|
5 |
4 |● (2,0)
3 |
2 |
1 |
0 |---------●--------- x
0 1 2 3 4
```
In a more detailed graph:
- The line \( y = -2x + 4 \) passes through \( (0, 4) \) and \( (2, 0) \).
- The line \( y = \frac{1}{2}x - 1 \) passes through \( (0, -1) \) and \( (2, 0) \).
- Both lines intersect at \( (2, 0) \), which is the solution to the system.
### Visual Graph
For a clearer visual, here's a graph created using [Desmos](https://www.desmos.com/calculator):
*(Please replace the image URL with an actual graph if needed.)*
## Conclusion
The system of equations:
\[
\begin{cases}
2x + y = 4 \\
y = \frac{1}{2}x - 1
\end{cases}
\]
has a single solution at the point \( (2, 0) \), where both lines intersect on the graph.
Reviewed and approved by the UpStudy tutoring team
Explain
Simplify this solution
