Question

Solve using elimination.

10x − 10y = 20
5x − 2y = –8

Answer 1

To solve the system of equations using elimination, we need to manipulate the equations such that we can eliminate one of the variables.

The system of equations is:

1. \( 10x - 10y = 20 \) (Equation 1)
2. \( 5x - 2y = -8 \) (Equation 2)

First, we can simplify Equation 1 by dividing every term by 10:

\[
x - y = 2 \quad \text{(Equation 3)}
\]

Now our equations are:

1. \( x - y = 2 \) (Equation 3)
2. \( 5x - 2y = -8 \) (Equation 2)

Next, we can multiply Equation 3 by 2 to make the coefficients of \( y \) equal:

\[
2(x - y) = 2 \cdot 2
\]

Which simplifies to:

\[
2x - 2y = 4 \quad \text{(Equation 4)}
\]

Now we have:

1. \( 2x - 2y = 4 \) (Equation 4)
2. \( 5x - 2y = -8 \) (Equation 2)

Now we can eliminate \( y \) by subtracting Equation 4 from Equation 2.

\[
(5x - 2y) - (2x - 2y) = -8 - 4
\]

This simplifies to:

\[
5x - 2y - 2x + 2y = -12
\]

Combining like terms gives us:

\[
3x = -12
\]

Now, divide by 3:

\[
x = -4
\]

Now that we've found \( x \), we can substitute it back into Equation 3 to find \( y \):

\[
x - y = 2
\]
\[
-4 - y = 2
\]

Solving for \( y \):

\[
-y = 2 + 4
\]
\[
-y = 6
\]
\[
y = -6
\]

Thus, the solution to the system of equations is:

\[
\boxed{(-4, -6)}
\]