Solve the following equation and verify the results.
`((3x+4))/8-((2x+5))/5=((3-4x))/10`

Answer

To solve the equation \(\frac{3x + 4}{8} - \frac{2x + 5}{5} = \frac{3 - 4x}{10}\), we will follow these steps:

### Step 1: Rewrite the equation
Start with the given equation:
\[ \frac{3x + 4}{8} - \frac{2x + 5}{5} = \frac{3 - 4x}{10} \]

### Step 2: Find a common denominator
The common denominator for 8, 5, and 10 is 40. We will multiply each term by 40 to eliminate the denominators:
\[ 40 \cdot \left(\frac{3x + 4}{8}\right) - 40 \cdot \left(\frac{2x + 5}{5}\right) = 40 \cdot \left(\frac{3 - 4x}{10}\right) \]

### Step 3: Simplify each term
Now, simplify each term:
\[ 5(3x + 4) - 8(2x + 5) = 4(3 - 4x) \]

### Step 4: Distribute
Distribute the numbers:
\[ 15x + 20 - 16x - 40 = 12 - 16x \]

### Step 5: Combine like terms
Combine like terms on the left side:
\[ -1x - 20 = 12 - 16x \]

### Step 6: Move all terms involving \(x\) to one side
Add \(16x\) to both sides:
\[ -1x + 16x - 20 = 12 \]
This simplifies to:
\[ 15x - 20 = 12 \]

### Step 7: Move the constant term to the other side
Add 20 to both sides:
\[ 15x = 32 \]

### Step 8: Solve for \(x\)
Divide both sides by 15:
\[ x = \frac{32}{15} \]

### Step 9: Verify the solution
Substitute \(x = \frac{32}{15}\) back into the original equation to verify:
1. Calculate the left side:
\[ \frac{3\left(\frac{32}{15}\right) + 4}{8} - \frac{2\left(\frac{32}{15}\right) + 5}{5} \]
Simplifying this gives:
\[ \frac{\frac{96}{15} + \frac{60}{15}}{8} - \frac{\frac{64}{15} + \frac{75}{15}}{5} \]
Which simplifies to:
\[ \frac{\frac{156}{15}}{8} - \frac{\frac{139}{15}}{5} \]

2. Calculate the right side:
\[ \frac{3 - 4\left(\frac{32}{15}\right)}{10} \]
Simplifying this gives:
\[ \frac{3 - \frac{128}{15}}{10} \]

If both sides are equal, then our solution is verified.