Question

Solve the following system of equations using the elimination method: 3x - 4y = -4, x + 3y = -10

Answer 1

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

1. \( 3x - 4y = -4 \) (Equation 1)
2. \( x + 3y = -10 \) (Equation 2)

The goal of elimination is to combine these equations to eliminate one of the variables. We can start by multiplying Equation 2 by 3 to align the coefficient of \( x \) in both equations.

Multiply Equation 2 by 3:

\[
3(x + 3y) = 3(-10)
\]

This simplifies to:

\[
3x + 9y = -30 \quad \text{(Equation 3)}
\]

Now we have:
1. \( 3x - 4y = -4 \) (Equation 1)
2. \( 3x + 9y = -30 \) (Equation 3)

Next, we can subtract Equation 1 from Equation 3 to eliminate \( 3x \):

\[
(3x + 9y) - (3x - 4y) = -30 - (-4)
\]

This simplifies to:

\[
9y + 4y = -30 + 4
\]
\[
13y = -26
\]

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

\[
y = -2
\]

Now that we have \( y \), we can substitute it back into either Equation 2 or Equation 1 to solve for \( x \). We'll use Equation 2 for substitution:

\[
x + 3(-2) = -10
\]
\[
x - 6 = -10
\]

Now, add 6 to both sides:

\[
x = -10 + 6
\]
\[
x = -4
\]

Thus, the solution for the system of equations is:

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

The ordered pair representing the solution to the system is:

\[
\boxed{(-4, -2)}
\]