To solve the system of linear equations:
[tex]\[
\begin{cases}
3x + 7y = 1 \\
y = x - 7
\end{cases}
\][/tex]
we'll use the substitution method given that the second equation is already solved for [tex]\( y \)[/tex]:
1. Substitute [tex]\( y = x - 7 \)[/tex] into the first equation [tex]\( 3x + 7y = 1 \)[/tex]:
[tex]\[
3x + 7(x - 7) = 1
\][/tex]
2. Simplify the equation to solve for [tex]\( x \)[/tex]:
[tex]\[
3x + 7x - 49 = 1
\][/tex]
Combine like terms:
[tex]\[
10x - 49 = 1
\][/tex]
Add 49 to both sides:
[tex]\[
10x = 50
\][/tex]
Divide both sides by 10:
[tex]\[
x = 5
\][/tex]
3. Substitute [tex]\( x = 5 \)[/tex] back into the second equation [tex]\( y = x - 7 \)[/tex] to solve for [tex]\( y \)[/tex]:
[tex]\[
y = 5 - 7
\][/tex]
[tex]\[
y = -2
\][/tex]
So, the solution to the system of equations is [tex]\( (5, -2) \)[/tex].
4. Verify which option matches this solution:
[tex]\[
\begin{array}{l}
A: (2, -5) \\
B: (5, -2) \\
C: (4, -3) \\
D: (3, -4)
\end{array}
\][/tex]
The correct answer is [tex]\( (5, -2) \)[/tex], which corresponds to.
[tex]\[
\boxed{B}
\][/tex]
