To solve the system of linear equations
[tex]\[
\begin{cases}
-7m + 4n = -20 \\
3m + n = 14
\end{cases}
\][/tex]
we need to find the values of [tex]\(m\)[/tex] and [tex]\(n\)[/tex] that satisfy both equations simultaneously. Here is a step-by-step solution:
1. Isolate one of the variables in one equation.
From the second equation, let's isolate [tex]\(n\)[/tex]:
[tex]\[
3m + n = 14 \quad \Rightarrow \quad n = 14 - 3m
\][/tex]
2. Substitute the expression for [tex]\(n\)[/tex] into the first equation.
Substitute [tex]\(n = 14 - 3m\)[/tex] into the first equation:
[tex]\[
-7m + 4(14 - 3m) = -20
\][/tex]
3. Simplify and solve for [tex]\(m\)[/tex].
First, distribute the 4:
[tex]\[
-7m + 56 - 12m = -20
\][/tex]
Combine the like terms:
[tex]\[
-19m + 56 = -20
\][/tex]
Next, isolate [tex]\(m\)[/tex] by subtracting 56 from both sides:
[tex]\[
-19m = -20 - 56
\][/tex]
Simplify the right-hand side:
[tex]\[
-19m = -76
\][/tex]
Solve for [tex]\(m\)[/tex] by dividing both sides by -19:
[tex]\[
m = \frac{-76}{-19} = 4
\][/tex]
4. Substitute the value of [tex]\(m\)[/tex] back into the expression for [tex]\(n\)[/tex].
Using the expression [tex]\(n = 14 - 3m\)[/tex]:
[tex]\[
n = 14 - 3(4) = 14 - 12 = 2
\][/tex]
5. Write the solution.
The solution to the system of equations is:
[tex]\[
m = 4, \quad n = 2
\][/tex]
So, the values of [tex]\(m\)[/tex] and [tex]\(n\)[/tex] that satisfy both equations are [tex]\(m = 4\)[/tex] and [tex]\(n = 2\)[/tex].
