Let’s solve the given system of equations step-by-step.
We start with two equations:
[tex]\[
\begin{array}{l}
y = 2 + 7x \quad (1) \\
-2x + 6y = 92 \quad (2)
\end{array}
\][/tex]
First, we'll use equation (1) to express [tex]\(y\)[/tex] in terms of [tex]\(x\)[/tex]:
[tex]\[
y = 2 + 7x
\][/tex]
Next, we'll substitute this expression for [tex]\(y\)[/tex] into equation (2):
[tex]\[
-2x + 6(2 + 7x) = 92
\][/tex]
Expanding the terms inside the parentheses:
[tex]\[
-2x + 12 + 42x = 92
\][/tex]
Combining like terms:
[tex]\[
-2x + 42x + 12 = 92
\][/tex]
Simplifying the left-hand side:
[tex]\[
40x + 12 = 92
\][/tex]
Next, we isolate [tex]\(x\)[/tex] by subtracting 12 from both sides:
[tex]\[
40x = 80
\][/tex]
Now divide both sides by 40:
[tex]\[
x = 2
\][/tex]
Having found [tex]\(x\)[/tex], we substitute it back into equation (1) to find [tex]\(y\)[/tex]:
[tex]\[
y = 2 + 7(2)
\][/tex]
Calculating inside the parentheses:
[tex]\[
y = 2 + 14
\][/tex]
So,
[tex]\[
y = 16
\][/tex]
Thus, the solution to the system of equations is [tex]\(x = 2\)[/tex] and [tex]\(y = 16\)[/tex].
Next, we check which of the given options matches this solution:
[tex]\[
\begin{array}{l}
x = -2, \, y = -12 \\
x = -4, \, y = 14 \\
x = 16, \, y = 2 \\
x = 2, \, y = 16 \\
\end{array}
\][/tex]
The matching option is:
[tex]\[
x = 2, \, y = 16
\][/tex]
Therefore, the solution to the system [tex]\((x, y)\)[/tex] that matches the given options is:
[tex]\[
(x, y) = (2, 16)
\][/tex]
