Okay, let's see. I need to find the sum of the values of 'a' that satisfy the equation (3)5² - 4(5 - a)² ÷ 3 = 63. Hmm, first, I should probably parse this equation step by step. Let me rewrite it to make sure I understand the operations.

So the equation is: 3 times 5 squared minus 4 times (5 minus a) squared divided by 3 equals 63. Wait, is the division only applied to the 4(5 - a)² term? Let me check the original equation again. It says "4(5 - a)^2 ÷ 3". So yes, the division by 3 applies to the entire term 4(5 - a)^2. So the equation is structured as [3 * 5²] - [4*(5 - a)² / 3] = 63.

Alright, let's break this down. Maybe I can simplify the left side step by step. First, compute 3 * 5². 5 squared is 25, so 3*25 is 75. Then the second term is [4*(5 - a)²]/3. So the equation becomes 75 - [4*(5 - a)²]/3 = 63.

To solve for 'a', I need to isolate the term with 'a'. Let me rearrange the equation. Subtract 75 from both sides: -[4*(5 - a)²]/3 = 63 - 75. That simplifies to -[4*(5 - a)²]/3 = -12. Then, multiply both sides by -1 to get [4*(5 - a)²]/3 = 12.

Now, multiply both sides by 3 to eliminate the denominator: 4*(5 - a)² = 36. Then divide both sides by 4: (5 - a)² = 9. Taking the square root of both sides, we get 5 - a = ±3. So, two cases:

Case 1: 5 - a = 3. Solving for 'a', subtract 5 from both sides: -a = 3 - 5 => -a = -2 => a = 2.

Case 2: 5 - a = -3. Similarly, subtract 5: -a = -3 -5 => -a = -8 => a = 8.

So the solutions are a = 2 and a = 8. The problem asks for the sum of these values. 2 + 8 = 10. Therefore, the answer should be 10.

Wait, let me double-check my steps to make sure I didn't make a mistake. Starting from the original equation:

3*5² - [4*(5 - a)²]/3 = 63

Calculating 3*25 is 75. Then 75 - [4*(5 - a)²]/3 = 63. Subtract 75: -[4*(5 - a)²]/3 = -12. Multiply by -1: [4*(5 - a)²]/3 = 12. Multiply by 3: 4*(5 - a)² = 36. Divide by 4: (5 - a)² = 9. Square root: 5 - a = ±3. So a = 5 ±3. That gives a = 5 + 3 = 8 and a = 5 - 3 = 2. Sum is 8 + 2 = 10. Yep, that seems right.

Alternatively, maybe I can verify by plugging the values back into the original equation.

First, let's try a = 2:

Left side: 3*5² - 4*(5 - 2)² /3 = 75 - 4*(3)² /3 = 75 - 4*9 /3 = 75 - 36/3 = 75 - 12 = 63. Which matches the right side.

Now a = 8:

Left side: 3*5² - 4*(5 - 8)² /3 = 75 - 4*(-3)² /3 = 75 - 4*9 /3 = 75 - 36/3 = 75 - 12 = 63. Also matches.

So both values satisfy the equation. Therefore, the sum is indeed 10. I think that's solid. No mistakes noticed here.
Therefore, the sum of the values of $a$ that satisfy the equation is $\boxed{10}$.
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