Certainly! Let's solve the system of equations using the elimination method step-by-step. We have the following system of linear equations:
[tex]\[
\begin{array}{l}
10m + 16n = 140 \quad \text{(1)} \\
5m - 8n = 60 \quad \text{(2)}
\end{array}
\][/tex]
Step-by-Step Solution:
1. First Equation (1):
[tex]\[ 10m + 16n = 140 \][/tex]
2. Second Equation (2):
[tex]\[ 5m - 8n = 60 \][/tex]
3. In order to eliminate one of the variables by addition or subtraction, let's align the coefficients for easier manipulation. Notice that if we multiply the second equation by 2, we will have the same coefficients for [tex]\(5m\)[/tex] that appear in the first equation:
Multiply Equation (2) by 2:
[tex]\[
2(5m - 8n) = 2 \cdot 60
\][/tex]
[tex]\[
10m - 16n = 120 \quad \text{(3)}
\][/tex]
4. Now we can add Equation (1) and Equation (3) to eliminate [tex]\(n\)[/tex]:
[tex]\[
(10m + 16n) + (10m - 16n) = 140 + 120
\][/tex]
[tex]\[
10m + 16n + 10m - 16n = 260
\][/tex]
[tex]\[
20m = 260
\][/tex]
5. Solving for [tex]\(m\)[/tex]:
[tex]\[
m = \frac{260}{20}
\][/tex]
[tex]\[
m = 13
\][/tex]
6. Substitute [tex]\(m = 13\)[/tex] back into either original equation to solve for [tex]\(n\)[/tex]. We use Equation (2) for this purpose:
[tex]\[
5m - 8n = 60
\][/tex]
[tex]\[
5(13) - 8n = 60
\][/tex]
[tex]\[
65 - 8n = 60
\][/tex]
[tex]\[
-8n = 60 - 65
\][/tex]
[tex]\[
-8n = -5
\][/tex]
[tex]\[
n = \frac{-5}{-8}
\][/tex]
[tex]\[
n = \frac{5}{8}
\][/tex]
7. Finally, the solution to the system of equations is:
[tex]\[
(m, n) = \left(13, \frac{5}{8}\right)
\][/tex]
Hence, the correct answer is [tex]\(\left(13, \frac{5}{8}\right)\)[/tex].
