Certainly! Let me provide a detailed, step-by-step solution to this problem.
We have a system of linear equations:
[tex]\[
\begin{array}{l}
m + n = 7 \quad \text{(Equation 1)} \\
2m - 3n = 4 \quad \text{(Equation 2)}
\end{array}
\][/tex]
To solve for [tex]\(m\)[/tex] and [tex]\(n\)[/tex], we can use either the substitution method or the elimination method. Here, we'll use the substitution method.
### Step 1: Solve one of the equations for one variable
From Equation 1, we can solve for [tex]\(m\)[/tex]:
[tex]\[ m + n = 7 \][/tex]
[tex]\[ m = 7 - n \][/tex]
### Step 2: Substitute the expression obtained into the other equation
Now, substitute [tex]\(m = 7 - n\)[/tex] into Equation 2:
[tex]\[ 2m - 3n = 4 \][/tex]
[tex]\[ 2(7 - n) - 3n = 4 \][/tex]
[tex]\[ 14 - 2n - 3n = 4 \][/tex]
[tex]\[ 14 - 5n = 4 \][/tex]
### Step 3: Solve for the variable
Isolate [tex]\(n\)[/tex]:
[tex]\[ 14 - 4 = 5n \][/tex]
[tex]\[ 10 = 5n \][/tex]
[tex]\[ n = \frac{10}{5} \][/tex]
[tex]\[ n = 2 \][/tex]
### Step 4: Substitute back to find the other variable
Now that we have [tex]\(n = 2\)[/tex], substitute it back into the expression for [tex]\(m\)[/tex] from Step 1:
[tex]\[ m = 7 - n \][/tex]
[tex]\[ m = 7 - 2 \][/tex]
[tex]\[ m = 5 \][/tex]
### Step 5: Calculate [tex]\(m - n\)[/tex]
To find the value of [tex]\(m - n\)[/tex]:
[tex]\[ m - n = 5 - 2 \][/tex]
[tex]\[ m - n = 3 \][/tex]
So, the value of [tex]\(m - n\)[/tex] is [tex]\( \boxed{3} \)[/tex].
