`3x-y=8-x`
`6x+4y=2y-9`
For the system of equations above, what is the value of the product xy?

Options

A. `-3`
B. `-2`
C. `2`
D. `3`

Answer

To solve the system of equations given by:

1. \( 3x - y = 8 - x \)
2. \( 6x + 4y = 2y - 9 \)

we will follow these steps:

### Step 1: Rearranging the first equation
We start with the first equation:
\[ 3x - y = 8 - x \]

We can rearrange this to bring all terms involving \(x\) and \(y\) to one side:
\[ 3x + x - y = 8 \]
\[ 4x - y = 8 \]

### Step 2: Rearranging the second equation
Now, we take the second equation:
\[ 6x + 4y = 2y - 9 \]

We can rearrange this as well:
\[ 6x + 4y - 2y = -9 \]
\[ 6x + 2y = -9 \]

### Step 3: Simplifying the equations
Now we have the simplified system of equations:
1. \( 4x - y = 8 \) (Equation 1)
2. \( 6x + 2y = -9 \) (Equation 2)

### Step 4: Multiplying and adding equations
To eliminate \(y\), we can multiply Equation 1 by 2:
\[ 2(4x - y) = 2(8) \]
This gives us:
\[ 8x - 2y = 16 \] (Equation 3)

Now we have:
1. \( 8x - 2y = 16 \) (Equation 3)
2. \( 6x + 2y = -9 \) (Equation 2)

### Step 5: Adding the equations
Now, we can add Equation 3 and Equation 2:
\[ (8x - 2y) + (6x + 2y) = 16 - 9 \]
This simplifies to:
\[ 14x = 7 \]

### Step 6: Solving for \(x\)
Now we can solve for \(x\):
\[ x = \frac{7}{14} = \frac{1}{2} \]

### Step 7: Finding \(y\)
Now that we have \(x\), we can substitute it back into one of the original equations to find \(y\). Let's use Equation 1:
\[ 4x - y = 8 \]
Substituting \(x = \frac{1}{2}\):
\[ 4 \left(\frac{1}{2}\right) - y = 8 \]
\[ 2 - y = 8 \]
\[ -y = 8 - 2 \]
\[ -y = 6 \]
\[ y = -6 \]

### Step 8: Finding the product \(xy\)
Now we can find the product \(xy\):
\[ xy = \left(\frac{1}{2}\right)(-6) = -3 \]

### Final Answer
The value of the product \(xy\) is \(-3\).