Solve using elimination.

5x + 8y = –3
–x + y = –15

1 answer

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:

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