Solve: $\frac{3 m+8}{8}=\frac{m-1}{3}$.

We are given the linear equation:
\[\frac{3m+8}{8} = \frac{m-1}{3}\]
Our objective is to isolate the variable $m$ and find its value. The equation involves two rational expressions, which can be simplified by eliminating the denominators. We identify the denominators as $8$ and $3$. To clear these, we find the least common multiple of $8$ and $3$, which is $24$, and multiply both sides of the equation by this value.

Clearing Denominators and Expanding

Multiplying both sides of the equation by $24$:
\[24 \times \left( \frac{3m+8}{8} \right) = 24 \times \left( \frac{m-1}{3} \right)\]
Simplifying the fractions on both sides:
\[3 \times (3m + 8) = 8 \times (m - 1)\]
Now, distribute the constants into the parentheses:
\[9m + 24 = 8m - 8\]

Solving for $m$

To isolate $m$, we group all terms involving $m$ on one side and the constant terms on the other. Subtract $8m$ from both sides:
\[9m - 8m + 24 = -8\]
\[m + 24 = -8\]
Next, subtract $24$ from both sides:
\[m = -8 - 24\]
\[m = -32\]

Verification and Final Result

To ensure the result is correct, substitute $m = -32$ back into the original equation:
\[\text{Left side: } \frac{3(-32)+8}{8} = \frac{-96+8}{8} = \frac{-88}{8} = -11\]
\[\text{Right side: } \frac{(-32)-1}{3} = \frac{-33}{3} = -11\]
Since both sides are equal, the value of $m$ is indeed $\boxed{-32}$.

Examples

This type of equation solving is fundamental in fields like physics or engineering, where ratios are often used to balance systems. For instance, if you are mixing chemical solutions or balancing mechanical levers, you might set up an equation where two different ratios must equate to a specific property. Solving for the unknown concentration or distance ensures the entire system behaves predictably and maintains equilibrium.