Solve the following equations.Check your result in each case
`(3x-1)/5-x/7=3`

Answer

To solve the equation \(\frac{3x - 1}{5} - \frac{x}{7} = 3\), we will follow these steps:

### Step 1: Find a common denominator
The denominators in the equation are 5 and 7. The least common multiple (LCM) of 5 and 7 is 35. We will multiply every term in the equation by 35 to eliminate the denominators.

\[ 35 \left(\frac{3x - 1}{5}\right) - 35 \left(\frac{x}{7}\right) = 35 \cdot 3 \]

### Step 2: Simplify each term
Now, we simplify each term:

- For the first term:
\[ 35 \cdot \frac{3x - 1}{5} = 7(3x - 1) = 21x - 7 \]

- For the second term:
\[ 35 \cdot \frac{x}{7} = 5x \]

- For the right side:
\[ 35 \cdot 3 = 105 \]

Putting it all together, we have:
\[ 21x - 7 - 5x = 105 \]

### Step 3: Combine like terms
Now, we will combine the \(x\) terms on the left side:
\[ (21x - 5x) - 7 = 105 \]
\[ 16x - 7 = 105 \]

### Step 4: Isolate the variable
Next, we will isolate \(x\) by adding 7 to both sides:
\[ 16x = 105 + 7 \]
\[ 16x = 112 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 16:
\[ x = \frac{112}{16} \]

### Step 6: Simplify the fraction
We can simplify \(\frac{112}{16}\) by dividing both the numerator and denominator by 16:
\[ x = 7 \]

### Step 7: Check the solution
To verify our solution, we will substitute \(x = 7\) back into the original equation:
\[ \frac{3(7) - 1}{5} - \frac{7}{7} = 3 \]
Calculating the left side:
\[ \frac{21 - 1}{5} - 1 = \frac{20}{5} - 1 = 4 - 1 = 3 \]

Since both sides are equal, our solution is verified.

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{7} \]