Let's solve the equation step by step:

Given equation:
\[
\frac{22}{5} \left(\frac{9}{7}m - \frac{1}{4}\right) = -\frac{503}{42} - \frac{5}{2}m
\]

### Step 1: Distribute \(\frac{22}{5}\) to the terms inside the parentheses.
\[
\frac{22}{5} \times \frac{9}{7}m - \frac{22}{5} \times \frac{1}{4} = -\frac{503}{42} - \frac{5}{2}m
\]
\[
\frac{198}{35}m - \frac{22}{20} = -\frac{503}{42} - \frac{5}{2}m
\]

### Step 2: Simplify the fraction \(\frac{22}{20}\).
\[
\frac{22}{20} = \frac{11}{10}
\]
So the equation becomes:
\[
\frac{198}{35}m - \frac{11}{10} = -\frac{503}{42} - \frac{5}{2}m
\]

### Step 3: Eliminate the fractions by finding a common denominator or multiply through by the least common multiple (LCM).
The LCM of 35, 10, 42, and 2 is 210.

Multiply the entire equation by 210 to eliminate the fractions:
\[
210 \times \left(\frac{198}{35}m - \frac{11}{10}\right) = 210 \times \left(-\frac{503}{42} - \frac{5}{2}m\right)
\]
\[
6 \times 198m - 21 \times 11 = 5 \times (-503) - 105 \times 5m
\]
\[
1188m - 231 = -2515 - 525m
\]

### Step 4: Combine like terms.
Add \(525m\) to both sides and add \(231\) to both sides:
\[
1188m + 525m = -2515 + 231
\]
\[
1713m = -2284
\]

### Step 5: Solve for \(m\) by dividing both sides by 1713.
\[
m = \frac{-2284}{1713}
\]
This fraction can be simplified if possible, but as it stands, this is the solution for \(m\).

Would you like any further details or a simplification of this fraction?

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**Related Questions:**
1. How do you solve equations with multiple fractions?
2. What is the method to find the least common multiple of different denominators?
3. Can you explain how to distribute a fraction over a subtraction inside parentheses?
4. How do you combine like terms in algebraic equations?
5. What is the process to simplify a fraction?

**Tip:** Always check if the final fraction can be simplified to its lowest terms to get the simplest form of the answer.

Learn how to solve the equation 22/5(9/7m - 1/4) = -503/42 - 5/2m step-by-step with detailed solutions. This problem involves algebraic manipulation, equations with fractions, and the distributive property. Suitable for high school students.

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