Solve the system of equations graphed on the coordinate axes below. $$ \begin{aligned} y & =-\frac{1}{2} x-6 \ y & =\frac{3}{2} x+2\end{aligned} $$

Question

UpStudy AI · Accepted Answer

To solve the system of equations:

$$
\begin{aligned}
y &= -\frac{1}{2}x - 6 \quad \text{(Equation 1)} \
y &= \frac{3}{2}x + 2 \quad \text{(Equation 2)}
\end{aligned}
$$

**Step 1: Set the equations equal to each other**

Since both equations equal $$ y $$, we can set the right-hand sides equal:

$$
-\frac{1}{2}x - 6 = \frac{3}{2}x + 2
$$

**Step 2: Solve for $$ x $$**

First, eliminate the fractions by multiplying every term by 2:

$$
- x - 12 = 3x + 4
$$

Now, bring all $$ x $$-terms to one side and constants to the other:

$$
- x - 3x = 4 + 12 \
-4x = 16 \
x = \frac{16}{-4} \
x = -4
$$

**Step 3: Solve for $$ y $$**

Substitute $$ x = -4 $$ into Equation 2:

$$
y = \frac{3}{2}(-4) + 2 \
y = -6 + 2 \
y = -4
$$

**Solution:**

The system of equations intersects at the point $$(-4, -4)$$.

$$\boxed{\,(-4,\ -4)\,}$$
The solution to the system of equations is the point $$(-4, -4)$$.

Solve the system of equations graphed on the coordinate axes below.