$$ \left. \begin{array} { l } { 6 x + 10 y = 0 } \ { - 6 x - 5 y = 15 } \ \hline \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { 6 x + 10 y = 0 } \ { - 6 x - 5 y = 15 } \ \hline \end{array} ight. $$

UpStudy AI · Accepted Answer

Let's solve the system of linear equations:

$$
\begin{cases}
6x + 10y = 0 \quad \text{(1)} \
-6x - 5y = 15 \quad \text{(2)}
\end{cases}
$$

**Step 1: Add Equations (1) and (2)**

Adding the two equations will help eliminate the variable $$ x $$:

$$
\begin{align*}
6x + 10y &= 0 \quad \text{(1)} \
-6x - 5y &= 15 \quad \text{(2)} \
\hline
(6x - 6x) + (10y - 5y) &= 0 + 15 \
5y &= 15
\end{align*}
$$

**Step 2: Solve for $$ y $$**

$$
5y = 15 \
y = \frac{15}{5} \
y = 3
$$

**Step 3: Substitute $$ y = 3 $$ into Equation (1) to Solve for $$ x $$**

$$
6x + 10(3) = 0 \
6x + 30 = 0 \
6x = -30 \
x = \frac{-30}{6} \
x = -5
$$

**Solution:**

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

You can verify the solution by plugging these values back into the original equations:

- **For Equation (1):**
  $$
  6(-5) + 10(3) = -30 + 30 = 0 \quad \checkmark
  $$

- **For Equation (2):**
  $$
  -6(-5) - 5(3) = 30 - 15 = 15 \quad \checkmark
  $$

Both equations are satisfied, confirming that $$ x = -5 $$ and $$ y = 3 $$ is the correct solution.
The solution to the system of equations is $$ x = -5 $$ and $$ y = 3 $$.

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