If ` 2 x + 3y = 12 and 3 x - 2y = 5 ` then

Options

A. ` x =2, y = 3 `
B. `x =2, y = - 3 `
C. `x = 3, y = 2 `
D. `x= 3, y = - 2 `

Answer

To solve the system of equations given by:

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

we will use the elimination method.

### Step 1: Multiply the equations to align coefficients
We want to eliminate one of the variables. Let's eliminate \( x \). To do this, we will multiply the first equation by 3 and the second equation by 2.

- Multiply Equation 1 by 3:
\[ 3(2x + 3y) = 3(12) \\ 6x + 9y = 36 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 2:
\[ 2(3x - 2y) = 2(5) \\ 6x - 4y = 10 \quad \text{(Equation 4)} \]

### Step 2: Subtract the equations
Now, we will subtract Equation 4 from Equation 3 to eliminate \( x \):

\[ (6x + 9y) - (6x - 4y) = 36 - 10 \\ 6x + 9y - 6x + 4y = 26 \\ 13y = 26 \]

### Step 3: Solve for \( y \)
Now, we can solve for \( y \):

\[ y = \frac{26}{13} = 2 \]

### Step 4: Substitute \( y \) back to find \( x \)
Now that we have \( y \), we can substitute it back into one of the original equations to find \( x \). Let's use Equation 1:

\[ 2x + 3(2) = 12 \\ 2x + 6 = 12 \]

### Step 5: Solve for \( x \)
Now, we solve for \( x \):

\[ 2x = 12 - 6 \\ 2x = 6 \\ x = \frac{6}{2} = 3 \]

### Final Solution
Thus, the solution to the system of equations is:
\[ x = 3, \quad y = 2 \]