The values of `x` and `y` in `3x+2y=4` and `2x-3y=7` are:

Options

A. `2, -1`
B. 2, 1
C. `-2, -1`
D. `-2, 1`

Answer

To solve the system of equations given by \(3x + 2y = 4\) and \(2x - 3y = 7\), we can use the method of elimination. Here’s a step-by-step solution:

### Step 1: Write the equations
We have two equations:
1. \(3x + 2y = 4\) (Equation 1)
2. \(2x - 3y = 7\) (Equation 2)

### Step 2: Multiply the equations to align coefficients
To eliminate \(y\), we can multiply Equation 1 by 3 and Equation 2 by 2:
- Multiply Equation 1 by 3:
\[ 3(3x + 2y) = 3(4) \implies 9x + 6y = 12 \quad (Equation 3) \]
- Multiply Equation 2 by 2:
\[ 2(2x - 3y) = 2(7) \implies 4x - 6y = 14 \quad (Equation 4) \]

### Step 3: Add the two new equations
Now, we will add Equation 3 and Equation 4:
\[ (9x + 6y) + (4x - 6y) = 12 + 14 \]
This simplifies to:
\[ 9x + 4x + 6y - 6y = 12 + 14 \implies 13x = 26 \]

### Step 4: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{26}{13} = 2 \]

### Step 5: Substitute \(x\) back into one of the original equations
Now that we have \(x\), we can substitute \(x = 2\) back into Equation 1 to find \(y\):
\[ 3(2) + 2y = 4 \]
This simplifies to:
\[ 6 + 2y = 4 \]
Subtract 6 from both sides:
\[ 2y = 4 - 6 \implies 2y = -2 \]

### Step 6: Solve for \(y\)
Now, divide by 2:
\[ y = \frac{-2}{2} = -1 \]

### Final Solution
Thus, the values of \(x\) and \(y\) are:
\[ x = 2, \quad y = -1 \]