1. Solve the following system by graphing. $$ \begin{array}{l}x+2 y=4 \ 3 x+y=-3\end{array} $$

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UpStudy AI · Accepted Answer

To solve the system of equations by graphing:

$$
\begin{cases}
x + 2y = 4 \
3x + y = -3
\end{cases}
$$

we'll find the equations of both lines in slope-intercept form (**y = mx + b**) and then determine their point of intersection.

### Step 1: Rewrite Both Equations in Slope-Intercept Form

**First Equation:**
$$
x + 2y = 4
$$
Solve for **y**:
$$
2y = -x + 4 \
y = -\frac{1}{2}x + 2
$$

**Second Equation:**
$$
3x + y = -3
$$
Solve for **y**:
$$
y = -3x - 3
$$

### Step 2: Graph Both Lines

**First Line ($$ y = -\frac{1}{2}x + 2 $$):**
- **Y-intercept (0, 2):** When $$ x = 0 $$, $$ y = 2 $$.
- **Slope (-1/2):** For every 2 units you move right on the x-axis, move 1 unit down on the y-axis.

**Second Line ($$ y = -3x - 3 $$):**
- **Y-intercept (0, -3):** When $$ x = 0 $$, $$ y = -3 $$.
- **Slope (-3):** For every 1 unit you move right on the x-axis, move 3 units down on the y-axis.

### Step 3: Find the Point of Intersection

To find where the two lines intersect, set the equations equal to each other:

$$
-\frac{1}{2}x + 2 = -3x - 3
$$

**Solve for x:**
$$
-\frac{1}{2}x + 3x = -3 - 2 \
\frac{5}{2}x = -5 \
x = -5 \times \frac{2}{5} \
x = -2
$$

**Now, solve for y using one of the equations:**
$$
y = -3(-2) - 3 \
y = 6 - 3 \
y = 3
$$

### Step 4: Solution

The two lines intersect at the point **(-2, 3)**. This is the solution to the system of equations.

### Visualization

If you were to graph the lines, they would cross at the point $$(-2, 3)$$, confirming that this is the unique solution to the system.

**Graph Representation:**

```
y
|
6 |          
5 |          
4 |     * (0,4)
3 |       \
2 | *     \
1 |        \
0 |---------*-------- x
-1|          \
-2|           \
-3|------------* (-2,3)
-4|
```

*(Note: This is a simplified text representation. For accurate graphing, using graph paper or a graphing tool/software is recommended.)*
The solution to the system is $$ x = -2 $$ and $$ y = 3 $$.

Solve the following system by graphing.

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