To solve the system of equations using elimination, let's follow a step-by-step approach:
### Given System of Equations:
[tex]\[
\begin{array}{c}
3g + 4h = 24 \\
-g + 2h = 2
\end{array}
\][/tex]
### Step 1: Eliminate One Variable
To eliminate the variable [tex]\( g \)[/tex], we will perform some operations on these equations. Let's start by eliminating [tex]\( g \)[/tex] from the second equation.
#### Adjust the Second Equation:
Multiply the entire second equation by 3 so that the coefficients of [tex]\( g \)[/tex] in both equations have the same magnitude but opposite signs:
[tex]\[
-3g + 6h = 6
\][/tex]
The system now becomes:
[tex]\[
\begin{array}{c}
3g + 4h = 24 \\
-3g + 6h = 6
\end{array}
\][/tex]
### Step 2: Add the Equations to Eliminate [tex]\( g \)[/tex]
Add the two equations together:
[tex]\[
(3g + 4h) + (-3g + 6h) = 24 + 6
\][/tex]
[tex]\[
3g - 3g + 4h + 6h = 30
\][/tex]
[tex]\[
0g + 10h = 30
\][/tex]
So, the resulting equation is:
[tex]\[
10h = 30
\][/tex]
### Step 3: Solve for [tex]\( h \)[/tex]
Divide both sides by 10:
[tex]\[
h = \frac{30}{10} = 3
\][/tex]
### Step 4: Substitute [tex]\( h \)[/tex] Back into One of the Original Equations
Now, we need to find the value of [tex]\( g \)[/tex]. Substitute [tex]\( h = 3 \)[/tex] into the second original equation [tex]\( -g + 2h = 2 \)[/tex]:
[tex]\[
-g + 2(3) = 2
\][/tex]
[tex]\[
-g + 6 = 2
\][/tex]
Subtract 6 from both sides:
[tex]\[
-g = 2 - 6
\][/tex]
[tex]\[
-g = -4
\][/tex]
Multiply both sides by -1:
[tex]\[
g = 4
\][/tex]
### Step 5: State the Solution
The solution to the system of equations is the ordered pair [tex]\( (g, h) \)[/tex]. Thus, in alphabetical order:
[tex]\[
(g, h) = (4, 3)
\][/tex]
The correct answer is:
[tex]\[
(4, 3)
\][/tex]
