If we set \( c = 37 \), the system of equations becomes:

1. \( a + b + 37 = 95 \)
2. \( a^2 + b^2 + 37^2 = 3083 \)

### Step 1: Solve for \( a + b \)
From the first equation:
\[
a + b = 95 - 37 = 58
\]

### Step 2: Solve for \( a^2 + b^2 \)
From the second equation:
\[
a^2 + b^2 + 37^2 = 3083
\]
First, calculate \( 37^2 \):
\[
37^2 = 1369
\]
So,
\[
a^2 + b^2 + 1369 = 3083
\]
Now solve for \( a^2 + b^2 \):
\[
a^2 + b^2 = 3083 - 1369 = 1714
\]

### Step 3: Use the identity for squares
We use the identity:
\[
(a + b)^2 = a^2 + b^2 + 2ab
\]
Substitute the known values:
\[
58^2 = 1714 + 2ab
\]
Calculate \( 58^2 \):
\[
3364 = 1714 + 2ab
\]
Solve for \( 2ab \):
\[
2ab = 3364 - 1714 = 1650
\]
So,
\[
ab = \frac{1650}{2} = 825
\]

### Step 4: Solve the quadratic equation
Now, \( a \) and \( b \) satisfy the quadratic equation:
\[
x^2 - (a+b)x + ab = 0
\]
Substitute \( a+b = 58 \) and \( ab = 825 \):
\[
x^2 - 58x + 825 = 0
\]

I'll now solve this quadratic equation to find \( a \) and \( b \).The solutions for the quadratic equation are:

- \( a = 25 \)
- \( b = 33 \)

Thus, the values of \( a \), \( b \), and \( c \) are:

- \( a = 25 \)
- \( b = 33 \)
- \( c = 37 \)

These values satisfy the given equations. 

If you have any further questions or need additional details, feel free to ask! Here are some related questions:

1. How do you verify solutions in systems of equations?
2. What happens when you substitute different values for \( c \)?
3. How can these solutions be interpreted geometrically?
4. What are the real-world applications of solving such equations?
5. Can you solve the system using a different method, like matrices?

**Tip:** Always double-check your final answers by substituting them back into the original equations to ensure correctness.

Learn how to solve a system of equations with a given constant c = 37. This problem involves algebraic manipulation, quadratic equations, and the identity for squares. Suitable for advanced high school students (Grades 10-12).

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