$$ \left. \begin{array} { c } { 15 , \frac { 3 } { 2 } x + 2 y = 12 } \ { x - 2 y = 6 } \end{array} ight. $$

Question

$$ \left. \begin{array} { c } { 15 , \frac { 3 } { 2 } x + 2 y = 12 } \ { x - 2 y = 6 } \end{array} ight. $$

UpStudy AI · Accepted Answer

It looks like you have a system of two linear equations:

$$
\begin{cases}
\displaystyle \frac{3}{2}x + 2y = 12 \
x - 2y = 6
\end{cases}
$$

Let's solve this system step by step.

### Step 1: Simplify the Equations

First, let's rewrite the equations for clarity:

1. $$\frac{3}{2}x + 2y = 12$$
2. $$x - 2y = 6$$

### Step 2: Solve for One Variable

We'll solve the second equation for $$x$$:

$$
x = 6 + 2y
$$

### Step 3: Substitute into the First Equation

Now, substitute $$x = 6 + 2y$$ into the first equation:

$$
\frac{3}{2}(6 + 2y) + 2y = 12
$$

Multiply out the terms:

$$
\frac{18}{2} + \frac{6}{2}y + 2y = 12
$$
$$
9 + 3y + 2y = 12
$$
$$
9 + 5y = 12
$$

### Step 4: Solve for $$y$$

Subtract 9 from both sides:

$$
5y = 3
$$
$$
y = \frac{3}{5}
$$

### Step 5: Solve for $$x$$

Now, substitute $$y = \frac{3}{5}$$ back into the expression for $$x$$:

$$
x = 6 + 2\left(\frac{3}{5}\right)
$$
$$
x = 6 + \frac{6}{5}
$$
$$
x = \frac{30}{5} + \frac{6}{5}
$$
$$
x = \frac{36}{5}
$$

### Final Solution

$$
x = \frac{36}{5} \quad \text{and} \quad y = \frac{3}{5}
$$

**In decimal form:**

$$
x = 7.2 \quad \text{and} \quad y = 0.6
$$
The solution to the system of equations is $$ x = \frac{36}{5} $$ and $$ y = \frac{3}{5} $$.

\( \left. \begin{array} { c } { 15 , \frac { 3 } { 2 } x + 2 y = 12 } \\ { x - 2 y = 6 } \end{array} \right. \)