Question

1. -x - 9y = 25

-x + 8y = -9

1. 6x + 2y = -8

10x + 2y = -8

Explanation:

Let's solve the system of equations for problem 12: \(

$$\begin{cases} 6x + 2y = -8 \\ 10x + 2y = -8 \end{cases}$$

\)

Step 1: Subtract the first equation from the second equation

Subtract \( 6x + 2y = -8 \) from \( 10x + 2y = -8 \) to eliminate \( y \).
\( (10x + 2y) - (6x + 2y) = -8 - (-8) \)
Simplify the left side: \( 10x + 2y - 6x - 2y = 4x \)
Simplify the right side: \( -8 + 8 = 0 \)
So we get \( 4x = 0 \)

Step 2: Solve for \( x \)

Divide both sides of \( 4x = 0 \) by 4:
\( x = \frac{0}{4} = 0 \)

Step 3: Substitute \( x = 0 \) into the first equation to solve for \( y \)

Substitute \( x = 0 \) into \( 6x + 2y = -8 \):
\( 6(0) + 2y = -8 \)
Simplify: \( 0 + 2y = -8 \)
\( 2y = -8 \)
Divide both sides by 2: \( y = \frac{-8}{2} = -4 \)

Step 1: Subtract the second equation from the first equation

Subtract \( -x + 8y = -9 \) from \( -x - 9y = 25 \) to eliminate \( x \).
\( (-x - 9y) - (-x + 8y) = 25 - (-9) \)
Simplify the left side: \( -x - 9y + x - 8y = -17y \)
Simplify the right side: \( 25 + 9 = 34 \)
So we get \( -17y = 34 \)

Step 2: Solve for \( y \)

Divide both sides of \( -17y = 34 \) by -17:
\( y = \frac{34}{-17} = -2 \)

Step 3: Substitute \( y = -2 \) into the first equation to solve for \( x \)

Substitute \( y = -2 \) into \( -x - 9y = 25 \):
\( -x - 9(-2) = 25 \)
Simplify: \( -x + 18 = 25 \)
Subtract 18 from both sides: \( -x = 25 - 18 = 7 \)
Multiply both sides by -1: \( x = -7 \)

Answer:

The solution to the system is \( x = 0 \), \( y = -4 \)

Now let's solve the system of equations for problem 10: \(

$$\begin{cases} -x - 9y = 25 \\ -x + 8y = -9 \end{cases}$$

\)