To solve the system of equations, we are given:
[tex]\[
\begin{array}{l}
y = 2 + 7x \\
-2x + 6y = 92
\end{array}
\][/tex]
Step 1: Substitute the expression for [tex]\( y \)[/tex] from the first equation into the second equation.
From the first equation, we have:
[tex]\[ y = 2 + 7x \][/tex]
Substitute this expression for [tex]\( y \)[/tex] into the second equation:
[tex]\[ -2x + 6(2 + 7x) = 92 \][/tex]
Step 2: Simplify and solve for [tex]\( x \)[/tex].
First, distribute the 6 in the equation:
[tex]\[ -2x + 6 \cdot 2 + 6 \cdot 7x = 92 \][/tex]
[tex]\[ -2x + 12 + 42x = 92 \][/tex]
Combine like terms:
[tex]\[ -2x + 42x + 12 = 92 \][/tex]
[tex]\[ 40x + 12 = 92 \][/tex]
Subtract 12 from both sides to isolate the term with [tex]\( x \)[/tex]:
[tex]\[ 40x = 80 \][/tex]
Divide by 40:
[tex]\[ x = 2 \][/tex]
Step 3: Substitute the value of [tex]\( x \)[/tex] back into the first equation to find [tex]\( y \)[/tex].
Using the first equation [tex]\( y = 2 + 7x \)[/tex], and substituting [tex]\( x = 2 \)[/tex]:
[tex]\[ y = 2 + 7(2) \][/tex]
[tex]\[ y = 2 + 14 \][/tex]
[tex]\[ y = 16 \][/tex]
Therefore, the solution to the system of equations is:
[tex]\[ x = 2 \][/tex]
[tex]\[ y = 16 \][/tex]
The given options are:
- [tex]\(x = -4, y = 14\)[/tex]
- [tex]\(x = 2, y = 16\)[/tex]
- [tex]\(x = -2, y = -12\)[/tex]
- [tex]\(x = 16, y = 2\)[/tex]
Thus, the correct solution is:
[tex]\[ x = 2, y = 16 \][/tex]
