Solve: `{:(3x + 5y = 12),(3x - 5y = - 18):}`

Answer

To solve the system of equations given by:

1) \(3x + 5y = 12\)
2) \(3x - 5y = -18\)

we will use the elimination method. Here are the steps to solve for \(x\) and \(y\):

### Step 1: Write down the equations
We have the two equations:
\[ 3x + 5y = 12 \quad \text{(Equation 1)} \]
\[ 3x - 5y = -18 \quad \text{(Equation 2)} \]

### Step 2: Eliminate one variable
Since the coefficients of \(x\) in both equations are the same, we can eliminate \(x\) by subtracting Equation 2 from Equation 1.

\[ (3x + 5y) - (3x - 5y) = 12 - (-18) \]

### Step 3: Simplify the equation
When we perform the subtraction, we get:
\[ 3x + 5y - 3x + 5y = 12 + 18 \]
This simplifies to:
\[ 10y = 30 \]

### Step 4: Solve for \(y\)
Now, divide both sides by 10:
\[ y = \frac{30}{10} = 3 \]

### Step 5: Substitute \(y\) back into one of the original equations
Now that we have \(y = 3\), we can substitute this value back into Equation 1 to find \(x\):
\[ 3x + 5(3) = 12 \]
This simplifies to:
\[ 3x + 15 = 12 \]

### Step 6: Solve for \(x\)
Subtract 15 from both sides:
\[ 3x = 12 - 15 \]
\[ 3x = -3 \]
Now, divide both sides by 3:
\[ x = \frac{-3}{3} = -1 \]

### Final Answer
Thus, the solution to the system of equations is:
\[ x = -1, \quad y = 3 \]

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