GG.8 Solve a system of equations using substitution

Question

J8X
Solve using substitution.

2x − 6y = –8
x − 4y = –8

Answer 1

To solve the system of equations using substitution, we have the following equations:

1. \( 2x - 6y = -8 \) (Equation 1)
2. \( x - 4y = -8 \) (Equation 2)

We can start with Equation 2 and solve for \( x \):

\[ x = 4y - 8 \]

Now that we have \( x \) in terms of \( y \), we can substitute this expression into Equation 1:

\[ 2(4y - 8) - 6y = -8 \]

Next, simplify and solve for \( y \):

1. Distribute \( 2 \) in the first term:

   \[ 8y - 16 - 6y = -8 \]

2. Combine like terms:

   \[ 2y - 16 = -8 \]

3. Add \( 16 \) to both sides:

   \[ 2y = 8 \]

4. Divide by \( 2 \):

   \[ y = 4 \]

Now that we have the value of \( y \), we substitute it back into the expression we found for \( x \):

\[ x = 4(4) - 8 \]

Calculate \( x \):

\[ x = 16 - 8 = 8 \]

Thus, the solution to the system of equations is:

\[ (x, y) = (8, 4) \]

To verify, plug the values back into the original equations:

1. For Equation 1: \[ 2(8) - 6(4) = 16 - 24 = -8 \quad \text{(True)} \]

2. For Equation 2: \[ 8 - 4(4) = 8 - 16 = -8 \quad \text{(True)} \]

Both equations hold true, confirming that the solution is correct.

The solution is:

\[ \boxed{(8, 4)} \]