Solve the equations and check your answer in the case:
`(3)/(4)(2x-5)-(5)/(6)(7-5x)=(7x)/(3)`

Answer

To solve the equation \( \frac{3}{4}(2x-5) - \frac{5}{6}(7-5x) = \frac{7x}{3} \), we will follow these steps:

### Step 1: Distribute the fractions
Distributing the fractions on the left side of the equation gives us:
\[ \frac{3}{4} \cdot 2x - \frac{3}{4} \cdot 5 - \frac{5}{6} \cdot 7 + \frac{5}{6} \cdot 5x = \frac{7x}{3} \]
This simplifies to:
\[ \frac{3 \cdot 2x}{4} - \frac{15}{4} - \frac{35}{6} + \frac{25x}{6} = \frac{7x}{3} \]

### Step 2: Simplify the fractions
Calculating the coefficients:
\[ \frac{3 \cdot 2x}{4} = \frac{6x}{4} = \frac{3x}{2} \]
Now, we have:
\[ \frac{3x}{2} - \frac{15}{4} - \frac{35}{6} + \frac{25x}{6} = \frac{7x}{3} \]

### Step 3: Combine like terms
To combine the \(x\) terms, we need a common denominator. The least common multiple of 2 and 6 is 6. Rewriting:
\[ \frac{3x}{2} = \frac{9x}{6} \]
Now, substituting back into the equation:
\[ \frac{9x}{6} + \frac{25x}{6} - \frac{15}{4} - \frac{35}{6} = \frac{7x}{3} \]
Combining the \(x\) terms:
\[ \frac{34x}{6} - \frac{15}{4} - \frac{35}{6} = \frac{7x}{3} \]

### Step 4: Move all \(x\) terms to one side
Now, let's move all \(x\) terms to one side:
\[ \frac{34x}{6} - \frac{7x}{3} = \frac{15}{4} + \frac{35}{6} \]

### Step 5: Find a common denominator for the fractions
The least common multiple of 6 and 3 is 6. Rewriting:
\[ \frac{7x}{3} = \frac{14x}{6} \]
Now, we have:
\[ \frac{34x}{6} - \frac{14x}{6} = \frac{15}{4} + \frac{35}{6} \]
This simplifies to:
\[ \frac{20x}{6} = \frac{15}{4} + \frac{35}{6} \]

### Step 6: Combine the constants on the right side
Finding a common denominator for \( \frac{15}{4} \) and \( \frac{35}{6} \):
The least common multiple of 4 and 6 is 12. Rewriting:
\[ \frac{15}{4} = \frac{45}{12}, \quad \frac{35}{6} = \frac{70}{12} \]
Now, we have:
\[ \frac{20x}{6} = \frac{45}{12} + \frac{70}{12} = \frac{115}{12} \]

### Step 7: Solve for \(x\)
Cross-multiplying gives:
\[ 20x \cdot 12 = 115 \cdot 6 \]
This simplifies to:
\[ 240x = 690 \]
Dividing both sides by 240:
\[ x = \frac{690}{240} = \frac{115}{40} = \frac{23}{8} \]

### Step 8: Final simplification
The value of \(x\) simplifies to:
\[ x = \frac{23}{40} \]

### Step 9: Check the solution
To check, substitute \(x = \frac{23}{40}\) back into the original equation and verify both sides are equal.