Solve the following equations and verify the results.
`2/5(4-2x)-3/4(2-5x)=7/5`

Answer

To solve the equation \( \frac{2}{5}(4 - 2x) - \frac{3}{4}(2 - 5x) = \frac{7}{5} \), we will follow these steps:

### Step 1: Distribute the fractions
Distributing the fractions on the left-hand side:
\[ \frac{2}{5} \cdot 4 - \frac{2}{5} \cdot 2x - \frac{3}{4} \cdot 2 + \frac{3}{4} \cdot 5x \]
Calculating each term:
\[ \frac{8}{5} - \frac{4}{5}x - \frac{6}{4} + \frac{15}{4}x \]
Now, simplify \(\frac{6}{4}\) to \(\frac{3}{2}\):
\[ \frac{8}{5} - \frac{4}{5}x - \frac{3}{2} + \frac{15}{4}x \]

### Step 2: Find a common denominator
The common denominator for \(5\) and \(4\) is \(20\). We will convert each term to have a denominator of \(20\):
\[ \frac{8 \cdot 4}{20} - \frac{4 \cdot 4}{20}x - \frac{3 \cdot 10}{20} + \frac{15 \cdot 5}{20}x \]
This simplifies to:
\[ \frac{32}{20} - \frac{16}{20}x - \frac{30}{20} + \frac{75}{20}x \]

### Step 3: Combine like terms
Combine the constant terms and the \(x\) terms:
\[ \left(\frac{32}{20} - \frac{30}{20}\right) + \left(-\frac{16}{20}x + \frac{75}{20}x\right) \]
This simplifies to:
\[ \frac{2}{20} + \frac{59}{20}x = \frac{7}{5} \]

### Step 4: Clear the fractions
Multiply the entire equation by \(20\) to eliminate the denominators:
\[ 2 + 59x = 28 \]

### Step 5: Solve for \(x\)
Subtract \(2\) from both sides:
\[ 59x = 26 \]
Now, divide by \(59\):
\[ x = \frac{26}{59} \]

### Step 6: Verification
Substituting \(x = \frac{26}{59}\) back into the original equation:
\[ \frac{2}{5}\left(4 - 2\left(\frac{26}{59}\right)\right) - \frac{3}{4}\left(2 - 5\left(\frac{26}{59}\right)\right) \]
Calculating \(4 - 2\left(\frac{26}{59}\right)\):
\[ 4 - \frac{52}{59} = \frac{236}{59} - \frac{52}{59} = \frac{184}{59} \]
Now calculate:
\[ \frac{2}{5} \cdot \frac{184}{59} = \frac{368}{295} \]
Now for \(2 - 5\left(\frac{26}{59}\right)\):
\[ 2 - \frac{130}{59} = \frac{118}{59} - \frac{130}{59} = -\frac{12}{59} \]
Calculating:
\[ -\frac{3}{4} \cdot -\frac{12}{59} = \frac{36}{236} = \frac{9}{59} \]
Now combine both parts:
\[ \frac{368}{295} + \frac{9}{59} \]
Finding a common denominator for \(295\) and \(59\) (which is \(295\)):
\[ \frac{368}{295} + \frac{45}{295} = \frac{413}{295} \]
Now simplify \(\frac{413}{295}\) to see if it equals \(\frac{7}{5}\):
\[ \frac{413}{295} = \frac{7 \cdot 59}{5 \cdot 59} = \frac{413}{295} \]
Thus, the left-hand side equals the right-hand side.

### Final Answer
The solution to the equation is:
\[ x = \frac{26}{59} \]