Given that `(y)/(sqrtx -3)= (sqrtx+3)/(3) and 2x + 42 =9x -63,` what is the value of y ?

Answer

To find the value of \( y \) given the equations

\[ \frac{y}{\sqrt{x} - 3} = \frac{\sqrt{x} + 3}{3} \]

and

\[ 2x + 42 = 9x - 63, \]

we will solve the equations step by step.

### Step 1: Solve for \( x \)

Start with the equation:

\[ 2x + 42 = 9x - 63. \]

Rearranging the equation, we move \( 2x \) to the right side and \( -63 \) to the left side:

\[ 42 + 63 = 9x - 2x. \]

This simplifies to:

\[ 105 = 7x. \]

Now, divide both sides by 7 to isolate \( x \):

\[ x = \frac{105}{7} = 15. \]

### Step 2: Substitute \( x \) into the first equation

Now that we have \( x = 15 \), we substitute this value into the first equation:

\[ \frac{y}{\sqrt{15} - 3} = \frac{\sqrt{15} + 3}{3}. \]

### Step 3: Simplify the equation

To eliminate the fraction, we cross-multiply:

\[ y \cdot 3 = (\sqrt{15} + 3)(\sqrt{15} - 3). \]

Using the difference of squares identity \( (a - b)(a + b) = a^2 - b^2 \), we can simplify the right side:

\[ y \cdot 3 = (\sqrt{15})^2 - 3^2 = 15 - 9 = 6. \]

### Step 4: Solve for \( y \)

Now, divide both sides by 3 to find \( y \):

\[ y = \frac{6}{3} = 2. \]

### Final Answer

Thus, the value of \( y \) is

\[ \boxed{2}. \]

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