graph the system of $$ 2 x+y=4 \quad $$ write the $$ \begin{array}{ll}y=\frac{1}{2} x-1 & ext { solution }\end{array} $$

Question

graph the system of $$ 2 x+y=4 \quad $$ write the $$ \begin{array}{ll}y=\frac{1}{2} x-1 & 	ext { solution }\end{array} $$

UpStudy AI · Accepted Answer

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):

![Graph of the system](https://i.imgur.com/yourimageurl.png)

*(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.
The system of equations has one solution at the point (2, 0).

graph the system of
write the

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