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

Answer

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

### Step 1: Move all terms involving \(x\) to one side of the equation.
We start with the equation:
\[ 3x - \frac{5}{3} = x - 3 \]
Subtract \(x\) from both sides:
\[ 3x - x - \frac{5}{3} = -3 \]
This simplifies to:
\[ 2x - \frac{5}{3} = -3 \]

### Step 2: Move the constant term to the other side.
Now, we add \(\frac{5}{3}\) to both sides:
\[ 2x = -3 + \frac{5}{3} \]
To perform this addition, we need a common denominator. The common denominator for 3 and 1 is 3. So, we rewrite \(-3\) as \(-\frac{9}{3}\):
\[ 2x = -\frac{9}{3} + \frac{5}{3} \]
Now, combine the fractions:
\[ 2x = \frac{-9 + 5}{3} = \frac{-4}{3} \]

### Step 3: Solve for \(x\).
Now, divide both sides by 2:
\[ x = \frac{-4}{3} \div 2 = \frac{-4}{3} \times \frac{1}{2} = \frac{-4}{6} = \frac{-2}{3} \]

### Step 4: Check the solution.
We need to verify that \(x = \frac{-2}{3}\) satisfies the original equation. We will substitute \(x\) back into the left and right sides of the equation.

**Left Side:**
\[ 3x - \frac{5}{3} = 3\left(\frac{-2}{3}\right) - \frac{5}{3} \]
Calculating this gives:
\[ = -2 - \frac{5}{3} \]
To combine these, we convert \(-2\) to a fraction with a denominator of 3:
\[ -2 = -\frac{6}{3} \]
So,
\[ -\frac{6}{3} - \frac{5}{3} = -\frac{11}{3} \]

**Right Side:**
\[ x - 3 = \frac{-2}{3} - 3 \]
Again, convert \(-3\) to a fraction:
\[ -3 = -\frac{9}{3} \]
So,
\[ \frac{-2}{3} - \frac{9}{3} = \frac{-11}{3} \]

Since both sides equal \(-\frac{11}{3}\), our solution is verified.

### Final Answer:
\[ x = \frac{-2}{3} \]