Sure! Let's solve the system of equations step by step to determine the correct ordered pair.
Consider the system of equations:
[tex]\[
\left\{
\begin{array}{l}
4x - 2y = 18 \\
-3x - 4y = -8
\end{array}
\right.
\][/tex]
We'll use the substitution or elimination method to solve this. The elimination method will be used here for detailed explanation.
1. Eliminate one variable:
- To eliminate [tex]\( y \)[/tex], we need to make the coefficients of [tex]\( y \)[/tex] in both equations equal (or opposites).
- Multiply the first equation by 2:
[tex]\[
2(4x - 2y) = 2(18)
\][/tex]
Which simplifies to:
[tex]\[
8x - 4y = 36 \quad \text{[Equation 3]}
\][/tex]
2. Combine the equations:
- Now we have the system:
[tex]\[
\left\{
\begin{array}{l}
8x - 4y = 36 \\
-3x - 4y = -8
\end{array}
\right.
\][/tex]
- Subtract the second equation from the first (to eliminate [tex]\( y \)[/tex]):
[tex]\[
(8x - 4y) - (-3x - 4y) = 36 - (-8)
\][/tex]
Which simplifies to:
[tex]\[
11x = 44
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{44}{11} = 4
\][/tex]
4. Substitute [tex]\( x = 4 \)[/tex] back into one of the original equations to solve for [tex]\( y \)[/tex]:
- Substitute [tex]\( x = 4 \)[/tex] into the first equation [tex]\( 4x - 2y = 18 \)[/tex]:
[tex]\[
4(4) - 2y = 18
\][/tex]
Simplifies to:
[tex]\[
16 - 2y = 18
\][/tex]
Subtract 16 from both sides:
[tex]\[
-2y = 2
\][/tex]
Divide by -2:
[tex]\[
y = -1
\][/tex]
Therefore, the solution to the system of equations is the ordered pair [tex]\( (4, -1) \)[/tex].
So, the correct answer is:
[tex]\[
(4, -1)
\][/tex]
