Question

Solve using elimination.

5x + 8y = –3
–x + y = –15

Answer 1

To solve the system of equations using elimination, we start with the following equations:

1. \( 5x + 8y = -3 \) (Equation 1)
2. \( -x + y = -15 \) (Equation 2)

To eliminate one of the variables, we'll first manipulate Equation 2 to align it with Equation 1. We can multiply Equation 2 by 5:

\[
5(-x + y) = 5(-15)
\]

This simplifies to:

\[
-5x + 5y = -75 \quad \text{(Equation 3)}
\]

Now we have the following system:

1. \( 5x + 8y = -3 \)
2. \( -5x + 5y = -75 \)

Next, we will add Equation 1 and Equation 3:

\[
(5x + 8y) + (-5x + 5y) = -3 + (-75)
\]

Simplifying the left side gives:

\[
0x + (8y + 5y) = -78
\]

which simplifies to:

\[
13y = -78
\]

Now divide both sides by 13 to solve for \( y \):

\[
y = \frac{-78}{13} = -6
\]

Now that we have \( y \), we can substitute it back into Equation 2 to solve for \( x \):

Substituting into Equation 2:

\[
-x + (-6) = -15
\]

This simplifies to:

\[
-x - 6 = -15
\]

Now add 6 to both sides:

\[
-x = -15 + 6
\]

This simplifies to:

\[
-x = -9
\]

Thus, multiplying both sides by -1, we get:

\[
x = 9
\]

Now we have the solution:

\[
x = 9 \quad \text{and } \quad y = -6
\]

To summarize, the solution to the system of equations is:

\[
(x, y) = (9, -6)
\]