Certainly! Let's solve the equation step by step.
Given the equation:
[tex]\[
\frac{m}{4} - \frac{m-1}{2} = \frac{1}{8} - \frac{m-2}{3}
\][/tex]
To eliminate the denominators, we'll multiply every term by 12, which is the least common multiple of 4, 2, 8, and 3.
Step 1: Multiply each term by 12:
[tex]\[
12 \left(\frac{m}{4}\right) - 12 \left(\frac{m-1}{2}\right) = 12 \left(\frac{1}{8}\right) - 12 \left(\frac{m-2}{3}\right)
\][/tex]
Step 2: Simplify each term:
[tex]\[
12 \cdot \frac{m}{4} = 3m
\][/tex]
[tex]\[
12 \cdot \frac{m-1}{2} = 6(m-1)
\][/tex]
[tex]\[
12 \cdot \frac{1}{8} = \frac{12}{8} = \frac{3}{2}
\][/tex]
[tex]\[
12 \cdot \frac{m-2}{3} = 4(m-2)
\][/tex]
Substituting these back into the equation, we get:
[tex]\[
3m - 6(m-1) = \frac{3}{2} - 4(m-2)
\][/tex]
Step 3: Expand and simplify the expressions:
On the left-hand side:
[tex]\[
3m - 6m + 6 = -3m + 6
\][/tex]
On the right-hand side:
[tex]\[
\frac{3}{2} - 4m + 8 = \frac{3}{2} - 4m + 8 = \frac{3}{2} + 8 - 4m = \frac{3}{2} + \frac{16}{2} - 4m = \frac{19}{2} - 4m
\][/tex]
So, we have:
[tex]\[
-3m + 6 = \frac{19}{2} - 4m
\][/tex]
Step 4: Isolate the variable [tex]\(m\)[/tex]:
To do this, we will add [tex]\(4m\)[/tex] to both sides:
[tex]\[
-3m + 4m + 6 = \frac{19}{2}
\][/tex]
This simplifies to:
[tex]\[
m + 6 = \frac{19}{2}
\][/tex]
Step 5: Solve for [tex]\(m\)[/tex]:
Subtract 6 from both sides to isolate [tex]\(m\)[/tex]:
[tex]\[
m = \frac{19}{2} - 6
\][/tex]
Convert 6 to a fraction with the same denominator:
[tex]\[
m = \frac{19}{2} - \frac{12}{2} = \frac{19 - 12}{2} = \frac{7}{2}
\][/tex]
Thus, the solution to the equation is:
[tex]\[
m = \frac{7}{2} = 3.5
\][/tex]
So the value of [tex]\(m\)[/tex] is:
[tex]\[ m = 3.5 \][/tex]
