9. Solve the system of equations using the method of your choice. $$ \begin{aligned} \frac{1}{3} x+y=1 \ 2 x+3 y=12\end{aligned} $$

Question

UpStudy AI · Accepted Answer

To solve the system of equations:

$$
\begin{aligned}
\frac{1}{3} x + y &= 1 \quad \quad \text{(Equation 1)} \
2x + 3y &= 12 \quad \quad \text{(Equation 2)}
\end{aligned}
$$

we can use the **substitution method**. Here's a step-by-step solution:

### Step 1: Solve Equation 1 for $$ y $$

Starting with Equation 1:
$$
\frac{1}{3} x + y = 1
$$

Subtract $$\frac{1}{3}x$$ from both sides to isolate $$ y $$:
$$
y = 1 - \frac{1}{3}x
$$

### Step 2: Substitute $$ y $$ into Equation 2

Now, substitute $$ y = 1 - \frac{1}{3}x $$ into Equation 2:
$$
2x + 3\left(1 - \frac{1}{3}x\right) = 12
$$

Simplify the equation:
$$
2x + 3 - x = 12
$$

Combine like terms:
$$
(2x - x) + 3 = 12 \
x + 3 = 12
$$

### Step 3: Solve for $$ x $$

Subtract 3 from both sides:
$$
x = 12 - 3 \
x = 9
$$

### Step 4: Find $$ y $$ Using the Value of $$ x $$

Substitute $$ x = 9 $$ back into the expression for $$ y $$:
$$
y = 1 - \frac{1}{3}(9) \
y = 1 - 3 \
y = -2
$$

### Final Solution

The solution to the system of equations is:
$$
x = 9, \quad y = -2
$$
The solution is $$ x = 9 $$ and $$ y = -2 $$.

Solve the system of equations using the
method of your choice.