Certainly! Let's follow a step-by-step approach to solve this system of linear equations using the elimination method.
The system of equations is:
[tex]\[
\begin{aligned}
1) \quad -2x + 4y &= 16 \\
2) \quad 2x + 2y &= 8
\end{aligned}
\][/tex]
Step 1: Eliminate one of the variables
To eliminate [tex]\( x \)[/tex], we can add the two equations together. This is because the coefficients of [tex]\( x \)[/tex] in the two equations are opposites ([tex]\(-2x\)[/tex] and [tex]\(2x\)[/tex]).
[tex]\[
(-2x + 4y) + (2x + 2y) = 16 + 8
\][/tex]
Step 2: Simplify the resulting equation
By combining like terms, we have:
[tex]\[
-2x + 2x + 4y + 2y = 16 + 8
\][/tex]
This simplifies to:
[tex]\[
6y = 24
\][/tex]
Step 3: Solve for [tex]\( y \)[/tex]
Divide both sides of the equation by 6:
[tex]\[
y = \frac{24}{6}
\][/tex]
[tex]\[
y = 4
\][/tex]
Step 4: Substitute [tex]\( y \)[/tex] back into one of the original equations
We can use either of the original equations to solve for [tex]\( x \)[/tex]. Let's use the second equation:
[tex]\[
2x + 2y = 8
\][/tex]
Substitute [tex]\( y = 4 \)[/tex]:
[tex]\[
2x + 2(4) = 8
\][/tex]
[tex]\[
2x + 8 = 8
\][/tex]
Step 5: Solve for [tex]\( x \)[/tex]
Subtract 8 from both sides:
[tex]\[
2x = 0
\][/tex]
Divide both sides by 2:
[tex]\[
x = 0
\][/tex]
Conclusion:
The solution to the system of equations is the ordered pair [tex]\((0, 4)\)[/tex].
By examining the list of options:
[tex]\((0,4)\)[/tex]
[tex]\((0,8)\)[/tex]
[tex]\((4,0)\)[/tex]
[tex]\((8,0)\)[/tex]
The correct solution is [tex]\((0, 4)\)[/tex].
