Solve the following equations.Check your result in each case
`(2x-1)/3-(6x-2)/5=1/3`

Options

A. `x=-9/2`
B. `x=-7/2`
C. `x=-1/2`
D. `x=-5/2`

Answer

To solve the equation \(\frac{2x-1}{3} - \frac{6x-2}{5} = \frac{1}{3}\), we will follow these steps:

### Step 1: Find the Least Common Multiple (LCM)
The denominators in the equation are 3 and 5. The LCM of 3 and 5 is 15.

### Step 2: Rewrite the Equation
Multiply each term by the LCM (15) to eliminate the denominators:
\[ 15 \left(\frac{2x-1}{3}\right) - 15 \left(\frac{6x-2}{5}\right) = 15 \left(\frac{1}{3}\right) \]

### Step 3: Simplify Each Term
This gives us:
\[ 5(2x - 1) - 3(6x - 2) = 5 \]

### Step 4: Distribute the Terms
Now distribute the 5 and -3:
\[ 10x - 5 - 18x + 6 = 5 \]

### Step 5: Combine Like Terms
Combine the \(x\) terms and the constant terms:
\[ (10x - 18x) + (-5 + 6) = 5 \]
This simplifies to:
\[ -8x + 1 = 5 \]

### Step 6: Isolate the Variable
Subtract 1 from both sides:
\[ -8x = 5 - 1 \]
\[ -8x = 4 \]

### Step 7: Solve for \(x\)
Divide both sides by -8:
\[ x = \frac{4}{-8} = -\frac{1}{2} \]

### Step 8: Check the Result
Now, we will substitute \(x = -\frac{1}{2}\) back into the original equation to check our solution:

Left-hand side:
\[ \frac{2(-\frac{1}{2}) - 1}{3} - \frac{6(-\frac{1}{2}) - 2}{5} \]
Calculating each part:
\[ = \frac{-1 - 1}{3} - \frac{-3 - 2}{5} \]
\[ = \frac{-2}{3} - \frac{-5}{5} \]
\[ = \frac{-2}{3} + 1 \]
Convert 1 to a fraction with a denominator of 3:
\[ = \frac{-2}{3} + \frac{3}{3} = \frac{1}{3} \]

Right-hand side:
\[ \frac{1}{3} \]

Since both sides are equal, our solution is verified.

### Final Answer:
\[ x = -\frac{1}{2} \]