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------------------------------------------------ Example 7

Solve the equation:

[tex]
\[
\frac{4m}{5} - \frac{2m}{3} = 4
\]
[/tex]

Solution:

The LCM of 5 and 3 is 15. Multiply both sides of the equation by 15, that is, multiply every term by 15:

[tex]
\[
\begin{aligned}
15 \times \frac{4m}{5} - 15 \times \frac{2m}{3} &= 15 \times 4 \\
3 \times 4m - 5 \times 2m &= 15 \times 4 \\
12m - 10m &= 60 \\
2m &= 60
\end{aligned}
\]
[/tex]

Divide both sides by 2:

[tex]
m = 30
[/tex]

Check:

When [tex] m = 30 [/tex]:

[tex]
\[
\begin{aligned}
\text{LHS} &= \frac{(4 \times 30)}{5} - \frac{(2 \times 30)}{3} \\
&= \frac{120}{5} - \frac{60}{3} \\
&= 24 - 20 \\
&= 4 = \text{RHS}
\end{aligned}
\]
[/tex]

To solve the equation

[tex]$$
\frac{4m}{5} - \frac{2m}{3} = 4,
$$[/tex]

follow these steps:

1. Determine the Least Common Multiple (LCM):
The denominators are 5 and 3. Their LCM is 15.

2. Multiply Each Term by 15:
Multiply every term of the equation by 15 to eliminate the fractions:

[tex]$$
15 \times \frac{4m}{5} - 15 \times \frac{2m}{3} = 15 \times 4.
$$[/tex]

Simplify each term:

- For the first term:
[tex]$$
15 \times \frac{4m}{5} = \left(\frac{15}{5}\right) \times 4m = 3 \times 4m = 12m.
$$[/tex]

- For the second term:
[tex]$$
15 \times \frac{2m}{3} = \left(\frac{15}{3}\right) \times 2m = 5 \times 2m = 10m.
$$[/tex]

- For the right-hand side:
[tex]$$
15 \times 4 = 60.
$$[/tex]

The equation now becomes:

[tex]$$
12m - 10m = 60.
$$[/tex]

3. Combine Like Terms and Solve for [tex]$m$[/tex]:
Combine the terms on the left-hand side:

[tex]$$
12m - 10m = 2m,
$$[/tex]

so the equation is:

[tex]$$
2m = 60.
$$[/tex]

Divide both sides by 2 to solve for [tex]$m$[/tex]:

[tex]$$
m = \frac{60}{2} = 30.
$$[/tex]

4. Check the Solution:
Substitute [tex]$m = 30$[/tex] back into the original equation:

[tex]$$
\frac{4(30)}{5} - \frac{2(30)}{3} = \frac{120}{5} - \frac{60}{3} = 24 - 20 = 4.
$$[/tex]

The left-hand side equals the right-hand side, confirming that [tex]$m = 30$[/tex] is the correct solution.

Thus, the solution to the equation is

[tex]$$
m = 30.
$$[/tex]