2x-3y=-14
3x-2y=-6
If (x ,y) is a solution to the system of equations above, what is the value of x − y ?

Options

A. `-20`
B. `-8`
C. `-4`
D. 8

Answer

To solve the system of equations and find the value of \( x - y \), we will follow these steps:

### Step 1: Write down the equations
We have the following two equations:
1. \( 2x - 3y = -14 \) (Equation 1)
2. \( 3x - 2y = -6 \) (Equation 2)

### Step 2: Eliminate one variable
To eliminate one of the variables, we can multiply Equation 1 by 3 and Equation 2 by 2. This will allow us to align the coefficients of \( x \) in both equations.

- Multiply Equation 1 by 3:
\[ 3(2x - 3y) = 3(-14) \implies 6x - 9y = -42 \quad \text{(Equation 3)} \]

- Multiply Equation 2 by 2:
\[ 2(3x - 2y) = 2(-6) \implies 6x - 4y = -12 \quad \text{(Equation 4)} \]

### Step 3: Subtract the equations
Now, we will subtract Equation 4 from Equation 3:
\[ (6x - 9y) - (6x - 4y) = -42 - (-12) \]
This simplifies to:
\[ -9y + 4y = -42 + 12 \]
\[ -5y = -30 \]

### Step 4: Solve for \( y \)
Now, we can solve for \( y \):
\[ y = \frac{-30}{-5} = 6 \]

### Step 5: Substitute \( y \) back to find \( x \)
Now that we have \( y = 6 \), we can substitute this value back into Equation 1 to find \( x \):
\[ 2x - 3(6) = -14 \]
\[ 2x - 18 = -14 \]
Adding 18 to both sides:
\[ 2x = -14 + 18 \]
\[ 2x = 4 \]
Dividing by 2:
\[ x = 2 \]

### Step 6: Calculate \( x - y \)
Now we can find \( x - y \):
\[ x - y = 2 - 6 = -4 \]

Thus, the value of \( x - y \) is \(-4\).

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