Question

Solve the system of equations using the linear combination method.

{6g+8h=40 −6g+2h=−20

Answer 1

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

1. \(6g + 8h = 40\) (Equation 1)
2. \(-6g + 2h = -20\) (Equation 2)

We can add both equations together to eliminate \(g\). In this case, the coefficients of \(g\) in both equations are opposites (6 and -6).

Adding the two equations:

\[
(6g - 6g) + (8h + 2h) = 40 - 20
\]

This simplifies to:

\[
0g + 10h = 20
\]

Thus, we get:

\[
10h = 20
\]

Now, solving for \(h\):

\[
h = \frac{20}{10} = 2
\]

Next, we substitute \(h = 2\) back into one of the original equations to solve for \(g\). We'll use Equation 1:

\[
6g + 8(2) = 40
\]

This becomes:

\[
6g + 16 = 40
\]

Subtracting 16 from both sides:

\[
6g = 40 - 16
\]

\[
6g = 24
\]

Now, solving for \(g\):

\[
g = \frac{24}{6} = 4
\]

We have the solutions:

\[
g = 4, \quad h = 2
\]

Thus, the solution to the system of equations is:

\[
(g, h) = (4, 2)
\]