Question

Solve the system of equations.
4x−8y=−36

−6x+y=32

Answer 1

To solve the system of equations:

1) \( 4x - 8y = -36 \) \\
2) \( -6x + y = 32 \)

We will first simplify the first equation and then solve the system.

### Step 1: Simplify the first equation

The first equation is \( 4x - 8y = -36 \).
We can simplify it by dividing every term by 4:

\[
x - 2y = -9
\]

### Step 2: Rewrite both equations

Now we will rewrite the system of equations:

1) \( x - 2y = -9 \) \\
2) \( -6x + y = 32 \)

### Step 3: Solve one of the equations for one variable

Let's solve the first equation for \( x \):

\[
x = 2y - 9
\]

### Step 4: Substitute into the second equation

Now substitute \( x \) in the second equation:

\[
-6(2y - 9) + y = 32
\]

### Step 5: Simplify and solve for \( y \)

Expanding this gives:

\[
-12y + 54 + y = 32
\]

Combine the \( y \) terms:

\[
-11y + 54 = 32
\]

Now, isolate \( y \):

\[
-11y = 32 - 54
\]
\[
-11y = -22
\]

Divide by -11:

\[
y = 2
\]

### Step 6: Substitute back to find \( x \)

Now plug \( y = 2 \) back into the equation we derived for \( x \):

\[
x = 2(2) - 9
\]
\[
x = 4 - 9
\]
\[
x = -5
\]

### Solution

So, the solution to the system of equations is:

\[
(x, y) = (-5, 2)
\]

### Step 7: Verification

To verify, we will substitute \( x = -5 \) and \( y = 2 \) back into the original equations:

1) \( 4(-5) - 8(2) = -20 - 16 = -36 \) (True) \\
2) \( -6(-5) + 2 = 30 + 2 = 32 \) (True)

Both equations are satisfied, thus the solution is correct.

The final answer is:

\[
\boxed{(-5, 2)}
\]