To find the solution to the system of equations:
[tex]\[
\begin{cases}
2x + 4y = 12 \\
y = \frac{1}{4}x - 3
\end{cases}
\][/tex]
we need to use substitution or elimination methods. Here, we will use substitution because the second equation is already solved for [tex]\(y\)[/tex].
First, substitute [tex]\(y = \frac{1}{4}x - 3\)[/tex] into the first equation:
[tex]\[
2x + 4\left(\frac{1}{4}x - 3\right) = 12
\][/tex]
Simplify the expression inside the parentheses:
[tex]\[
2x + 4\left(\frac{1}{4}x\right) - 4 \cdot 3 = 12
\][/tex]
This becomes:
[tex]\[
2x + x - 12 = 12
\][/tex]
Combine like terms:
[tex]\[
3x - 12 = 12
\][/tex]
Add 12 to both sides to isolate the term with [tex]\(x\)[/tex]:
[tex]\[
3x = 24
\][/tex]
Divide both sides by 3:
[tex]\[
x = 8
\][/tex]
Now that we have [tex]\(x\)[/tex], substitute [tex]\(x = 8\)[/tex] back into the second equation, [tex]\(y = \frac{1}{4}x - 3\)[/tex]:
[tex]\[
y = \frac{1}{4}(8) - 3
\][/tex]
Simplify the right side:
[tex]\[
y = 2 - 3
\][/tex]
Thus, we get:
[tex]\[
y = -1
\][/tex]
So the solution to the system is [tex]\( (8, -1) \)[/tex].
The correct answer is:
[tex]\[ (8, -1) \][/tex]
