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Solve the following system of equations using the elimination method. 3x−4y=−4 , x+3y=−10 (1 point) Responses (−16,2) left parenthesis negative 16 comma 2 right parenthesis (−4,−2) left parenthesis negative 4 comma negative 2 right parenthesis (−16,−2) left parenthesis negative 16 comma negative 2 right parenthesis (−2,−4) left parenthesis negative 2 comma negative 4 right parenthesis

1 answer

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

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

First, we can eliminate one of the variables. Here, we will eliminate \( x \) by manipulating the equations.

Step 1: Make the coefficients of \( x \) the same.

To match the coefficients of \( x \) in both equations, we can multiply Equation 2 by 3:

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

This gives us:

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

Now we have:

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

Step 2: Subtract Equation 1 from Equation 3.

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

This simplifies to:

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

Thus, we have:

\[ 13y = -26 \]

Step 3: Solve for \( y \).

\[ y = \frac{-26}{13} = -2 \]

Step 4: Substitute \( y \) back into one of the original equations to find \( x \). We can use Equation 2:

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

This becomes:

\[ x - 6 = -10 \]

Solving for \( x \):

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

Thus, the solution to the system of equations is:

\[ (x, y) = (-4, -2) \]

Therefore, the answer is \( \boxed{(-4, -2)} \).