Let's solve the system of equations using substitution:
[tex]\[
\begin{cases}
4x + 2y = 6 \\
x = 2y + 4
\end{cases}
\][/tex]
Step 1: Substitute [tex]\( x = 2y + 4 \)[/tex] into the first equation.
The first equation is:
[tex]\[
4x + 2y = 6.
\][/tex]
Replace [tex]\( x \)[/tex] with [tex]\( 2y + 4 \)[/tex]:
[tex]\[
4(2y + 4) + 2y = 6.
\][/tex]
Step 2: Simplify and solve for [tex]\( y \)[/tex].
Distribute the [tex]\( 4 \)[/tex]:
[tex]\[
8y + 16 + 2y = 6.
\][/tex]
Combine like terms:
[tex]\[
10y + 16 = 6.
\][/tex]
Isolate [tex]\( y \)[/tex] by subtracting 16 from both sides:
[tex]\[
10y = 6 - 16,
\][/tex]
[tex]\[
10y = -10.
\][/tex]
Divide both sides by 10:
[tex]\[
y = -1.
\][/tex]
Step 3: Solve for [tex]\( x \)[/tex].
Substitute [tex]\( y = -1 \)[/tex] back into the second equation:
[tex]\[
x = 2y + 4.
\][/tex]
So,
[tex]\[
x = 2(-1) + 4,
\][/tex]
[tex]\[
x = -2 + 4,
\][/tex]
[tex]\[
x = 2.
\][/tex]
Step 4: Write the solution as an ordered pair.
The solution to the system is:
[tex]\[
(x, y) = (2, -1).
\][/tex]
Step 5: Verify the solution.
Check the solution [tex]\((2, -1)\)[/tex] in both original equations:
For the first equation:
[tex]\[
4(2) + 2(-1) = 6,
\][/tex]
[tex]\[
8 - 2 = 6,
\][/tex]
[tex]\[
6 = 6 \quad \text{(True)}.
\][/tex]
For the second equation:
[tex]\[
2 = 2(-1) + 4,
\][/tex]
[tex]\[
2 = -2 + 4,
\][/tex]
[tex]\[
2 = 2 \quad \text{(True)}.
\][/tex]
Thus, the solution [tex]\((2, -1)\)[/tex] satisfies both equations. The correct answer from the provided options is:
[tex]\[
(2, -1).
\][/tex]
