To solve the system of linear equations and find the value of [tex]\( x - y \)[/tex], follow these steps:
1. Consider the given system of equations:
[tex]\[
7x + y = 23 \quad \text{(Equation 1)}
\][/tex]
[tex]\[
x + 7y = 47 \quad \text{(Equation 2)}
\][/tex]
2. Our goal is to determine the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. Begin by expressing one of the variables in terms of the other. Let’s solve Equation 1 for [tex]\( y \)[/tex]:
[tex]\[
y = 23 - 7x
\][/tex]
3. Substitute this expression for [tex]\( y \)[/tex] into Equation 2 to eliminate [tex]\( y \)[/tex]:
[tex]\[
x + 7(23 - 7x) = 47
\][/tex]
4. Distribute the 7 within the parentheses:
[tex]\[
x + 161 - 49x = 47
\][/tex]
5. Combine like terms:
[tex]\[
x - 49x + 161 = 47
\][/tex]
[tex]\[
-48x + 161 = 47
\][/tex]
6. Isolate [tex]\( x \)[/tex] by first subtracting 161 from both sides of the equation:
[tex]\[
-48x = 47 - 161
\][/tex]
[tex]\[
-48x = -114
\][/tex]
7. Divide both sides by -48 to solve for [tex]\( x \)[/tex]:
[tex]\[
x = \frac{-114}{-48} = \frac{114}{48} = \frac{19}{8}
\][/tex]
8. Next, substitute [tex]\( x = \frac{19}{8} \)[/tex] back into the expression for [tex]\( y \)[/tex] derived from Equation 1:
[tex]\[
y = 23 - 7 \left( \frac{19}{8} \right)
\][/tex]
[tex]\[
y = 23 - \frac{133}{8}
\][/tex]
9. To simplify, first convert 23 to a fraction with a denominator of 8:
[tex]\[
y = \frac{184}{8} - \frac{133}{8}
\][/tex]
10. Subtract the numerators:
[tex]\[
y = \frac{184 - 133}{8} = \frac{51}{8}
\][/tex]
11. Now that we have [tex]\( x = \frac{19}{8} \)[/tex] and [tex]\( y = \frac{51}{8} \)[/tex], we can find [tex]\( x - y \)[/tex]:
[tex]\[
x - y = \frac{19}{8} - \frac{51}{8} = \frac{19 - 51}{8} = \frac{-32}{8} = -4
\][/tex]
Therefore, the value of [tex]\( x - y \)[/tex] is [tex]\(\boxed{-4}\)[/tex]. So, the correct answer is:
[tex]\[B) -4\][/tex]
