$$ 3 x+4 y=-18 $$ and $$ 5 x-2 y=-4 $$

Question

UpStudy AI · Accepted Answer

To solve the system of linear equations:

$$
\begin{cases}
3x + 4y = -18 \quad \text{(1)} \
5x - 2y = -4 \quad \text{(2)}
\end{cases}
$$

we can use the **elimination method** to find the values of $$ x $$ and $$ y $$.

### Step 1: Make the Coefficients of One Variable Opposites

Let's eliminate $$ y $$ by making the coefficients of $$ y $$ in both equations opposites.

- **Equation (1):** $$ 3x + 4y = -18 $$
- **Equation (2):** $$ 5x - 2y = -4 $$

To make the coefficients of $$ y $$ opposites, multiply **Equation (2)** by **2**:

$$
2 \times (5x - 2y) = 2 \times (-4) \
10x - 4y = -8 \quad \text{(3)}
$$

Now, the system looks like:

$$
\begin{cases}
3x + 4y = -18 \quad \text{(1)} \
10x - 4y = -8 \quad \text{(3)}
\end{cases}
$$

### Step 2: Add the Equations to Eliminate $$ y $$

Add **Equation (1)** and **Equation (3)**:

$$
(3x + 4y) + (10x - 4y) = -18 + (-8) \
13x = -26
$$

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

$$
x = \frac{-26}{13} \
x = -2
$$

### Step 4: Substitute $$ x $$ into One of the Original Equations to Find $$ y $$

Let's substitute $$ x = -2 $$ into **Equation (1)**:

$$
3(-2) + 4y = -18 \
-6 + 4y = -18 \
4y = -18 + 6 \
4y = -12 \
y = \frac{-12}{4} \
y = -3
$$

### Solution

$$
x = -2 \
y = -3
$$

**Therefore, the solution to the system is $$ x = -2 $$ and $$ y = -3 $$.**
The solution is $$ x = -2 $$ and $$ y = -3 $$.

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