Solve the following eqations.Check your result in each case
`2X-1/3=1/5-X`

Answer

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

### Step 1: Rearranging the Equation
We want to get all terms involving \(x\) on one side and constant terms on the other side. Let's add \(x\) to both sides of the equation.

\[ 2x + x - \frac{1}{3} = \frac{1}{5} \]

This simplifies to:

\[ 3x - \frac{1}{3} = \frac{1}{5} \]

**Hint:** To isolate \(x\), you can move the \(x\) terms to one side and the constant terms to the other side.

### Step 2: Adding \(\frac{1}{3}\) to Both Sides
Next, we will add \(\frac{1}{3}\) to both sides to eliminate the constant term on the left side.

\[ 3x = \frac{1}{5} + \frac{1}{3} \]

**Hint:** When adding fractions, make sure to find a common denominator.

### Step 3: Finding a Common Denominator
The least common multiple of 5 and 3 is 15. We will convert both fractions to have a denominator of 15.

\[ \frac{1}{5} = \frac{3}{15} \quad \text{and} \quad \frac{1}{3} = \frac{5}{15} \]

So we can rewrite the equation as:

\[ 3x = \frac{3}{15} + \frac{5}{15} \]

**Hint:** Always convert fractions to the same denominator before adding them.

### Step 4: Adding the Fractions
Now, we can add the fractions on the right side:

\[ 3x = \frac{3 + 5}{15} = \frac{8}{15} \]

**Hint:** Combine the numerators while keeping the denominator the same.

### Step 5: Solving for \(x\)
Now, we divide both sides by 3 to solve for \(x\):

\[ x = \frac{8}{15} \div 3 = \frac{8}{15} \times \frac{1}{3} = \frac{8}{45} \]

**Hint:** Dividing by a number is the same as multiplying by its reciprocal.

### Step 6: Checking the Result
To verify our solution, we will substitute \(x = \frac{8}{45}\) back into the original equation:

Left Side:
\[ 2x - \frac{1}{3} = 2 \times \frac{8}{45} - \frac{1}{3} = \frac{16}{45} - \frac{1}{3} \]
Convert \(\frac{1}{3}\) to have a denominator of 45:
\[ \frac{1}{3} = \frac{15}{45} \]
So,
\[ \frac{16}{45} - \frac{15}{45} = \frac{1}{45} \]

Right Side:
\[ \frac{1}{5} - x = \frac{1}{5} - \frac{8}{45} \]
Convert \(\frac{1}{5}\) to have a denominator of 45:
\[ \frac{1}{5} = \frac{9}{45} \]
So,
\[ \frac{9}{45} - \frac{8}{45} = \frac{1}{45} \]

Since both sides are equal, the solution is verified.

**Final Answer:**
\[ x = \frac{8}{45} \]