If `4x - 5y =-37 and 7x + y =- 16,` then `x - y=`____

Options

A. `-8`
B. 8
C. 2
D. `-2`

Answer

To solve the system of equations given by:

1. \( 4x - 5y = -37 \) (Equation 1)
2. \( 7x + y = -16 \) (Equation 2)

we will follow these steps:

### Step 1: Multiply Equation 2 by 5
We multiply Equation 2 by 5 to align the coefficients of \(y\) for elimination.

\[ 5(7x + y) = 5(-16) \]

This gives us:

\[ 35x + 5y = -80 \quad \text{(Equation 3)} \]

### Step 2: Write down the modified equations
Now we have:

- Equation 1: \( 4x - 5y = -37 \)
- Equation 3: \( 35x + 5y = -80 \)

### Step 3: Add Equation 1 and Equation 3
Next, we add Equation 1 and Equation 3 to eliminate \(y\):

\[ (4x - 5y) + (35x + 5y) = -37 - 80 \]

This simplifies to:

\[ 39x = -117 \]

### Step 4: Solve for \(x\)
Now, we solve for \(x\):

\[ x = \frac{-117}{39} = -3 \]

### Step 5: Substitute \(x\) back into Equation 2 to find \(y\)
Now, we substitute \(x = -3\) back into Equation 2 to find \(y\):

\[ 7(-3) + y = -16 \]

This simplifies to:

\[ -21 + y = -16 \]

Adding 21 to both sides gives:

\[ y = 5 \]

### Step 6: Find \(x - y\)
Now that we have both \(x\) and \(y\), we can find \(x - y\):

\[ x - y = -3 - 5 = -8 \]

Thus, the final answer is:

\[ \boxed{-8} \]

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