1. (by hand) $$ \left\{\begin{array}{l}3 x+2 y=19 \ 3 x-5 y=-9\end{array} ight. $$

Question

1. (by hand) $$ \left\{\begin{array}{l}3 x+2 y=19 \ 3 x-5 y=-9\end{array}ight. $$

UpStudy AI · Accepted Answer

To solve the system of linear equations:

$$
\begin{cases}
3x + 2y = 19 \quad \text{(1)} \
3x - 5y = -9 \quad \text{(2)}
\end{cases}
$$

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

### Step 1: Subtract Equation (2) from Equation (1)

By eliminating one of the variables, we can solve for the other. Firstly, subtract equation (2) from equation (1):

$$
\begin{align*}
(3x + 2y) - (3x - 5y) &= 19 - (-9) \
3x + 2y - 3x + 5y &= 19 + 9 \
\end{align*}
$$

Simplify the equation:

$$
7y = 28
$$

### Step 2: Solve for $$ y $$

$$
y = \frac{28}{7} = 4
$$

### Step 3: Substitute $$ y = 4 $$ into One of the Original Equations

Let's substitute $$ y = 4 $$ into equation (1):

$$
3x + 2(4) = 19 \
3x + 8 = 19
$$

Subtract 8 from both sides:

$$
3x = 11
$$

### Step 4: Solve for $$ x $$

$$
x = \frac{11}{3}
$$

### Final Solution

$$
x = \frac{11}{3}, \quad y = 4
$$

So, the solution to the system of equations is:

$$
\boxed{\left( \dfrac{11}{3},\ 4 \right)}
$$
The solution is $$ x = \frac{11}{3} $$ and $$ y = 4 $$.

1. (by hand) \( \left\{\begin{array}{l}3 x+2 y=19 \\ 3 x-5 y=-9\end{array}\right. \)