If `((x)/(3)+1 ,y-(2)/(3))=((5)/(3),(1)/(3))`, then find the values of x and y.

Answer

To solve the equation \(\left(\frac{x}{3} + 1, y - \frac{2}{3}\right) = \left(\frac{5}{3}, \frac{1}{3}\right)\), we can equate the corresponding components of the ordered pairs.

### Step 1: Equate the first components
From the first components of the ordered pairs, we have:
\[ \frac{x}{3} + 1 = \frac{5}{3} \]

### Step 2: Isolate \(\frac{x}{3}\)
To isolate \(\frac{x}{3}\), we subtract 1 from both sides:
\[ \frac{x}{3} = \frac{5}{3} - 1 \]
Converting 1 to a fraction with a denominator of 3:
\[ 1 = \frac{3}{3} \]
So,
\[ \frac{x}{3} = \frac{5}{3} - \frac{3}{3} = \frac{2}{3} \]

### Step 3: Solve for \(x\)
Now, multiply both sides by 3 to solve for \(x\):
\[ x = 3 \times \frac{2}{3} = 2 \]

### Step 4: Equate the second components
Now, we equate the second components of the ordered pairs:
\[ y - \frac{2}{3} = \frac{1}{3} \]

### Step 5: Isolate \(y\)
To isolate \(y\), we add \(\frac{2}{3}\) to both sides:
\[ y = \frac{1}{3} + \frac{2}{3} \]

### Step 6: Solve for \(y\)
Now, add the fractions:
\[ y = \frac{1 + 2}{3} = \frac{3}{3} = 1 \]

### Final Values
Thus, the values of \(x\) and \(y\) are:
\[ x = 2, \quad y = 1 \]