We are given the following system of equations:

1. $a + b + c = 24$
2. $\frac{ab}{2} = 24$, which simplifies to $ab = 48$.

To solve for $a$, $b$, and $c$, let's proceed step by step.

Step 1: Express $ab$ in terms of a single variable

From the second equation: $ab = 48.$ This means $a$ and $b$ are factors of 48. Possible pairs $(a, b)$ can be:

• $(1, 48)$
• $(2, 24)$
• $(3, 16)$
• $(4, 12)$
• $(6, 8)$
• $(8, 6)$
• $(12, 4)$
• $(16, 3)$
• $(24, 2)$
• $(48, 1)$

Step 2: Use the sum equation

From the first equation: $a + b + c = 24,$ we can solve for $c$: $c = 24 - (a + b).$ Now, we substitute the values of $a$ and $b$ from the factor pairs into this equation to find $c$.

Step 3: Check for valid solutions

Let's check each pair:

1. For $a = 1$, $b = 48$, we get: $c = 24 - (1 + 48) = 24 - 49 = -25.$ This gives a negative value for $c$, which may not be valid in this context.

2. For $a = 2$, $b = 24$, we get: $c = 24 - (2 + 24) = 24 - 26 = -2.$ Again, $c$ is negative.

3. For $a = 3$, $b = 16$, we get: $c = 24 - (3 + 16) = 24 - 19 = 5.$ This gives a positive value for $c$, and is a valid solution.

4. For $a = 4$, $b = 12$, we get: $c = 24 - (4 + 12) = 24 - 16 = 8.$ This is also a valid solution.

5. For $a = 6$, $b = 8$, we get: $c = 24 - (6 + 8) = 24 - 14 = 10.$ This is another valid solution.

Thus, we have three valid solutions:

• $a = 3$, $b = 16$, $c = 5$,
• $a = 4$, $b = 12$, $c = 8$,
• $a = 6$, $b = 8$, $c = 10$.

Step 4: Final solutions

The possible sets of values for $a$, $b$, and $c$ are:

1. $a = 3$, $b = 16$, $c = 5$,
2. $a = 4$, $b = 12$, $c = 8$,
3. $a = 6$, $b = 8$, $c = 10$.

Would you like more details or further clarification on any of these steps? Here are a few follow-up questions to help:

1. How do you check the consistency of multiple solutions in systems of equations?
2. What are some real-life applications of solving systems like these?
3. How can you extend this type of problem to include more variables?
4. What would happen if one of the numbers were constrained to be negative?
5. How do you recognize when a system has multiple valid solutions?

Tip: When dealing with factorizations like $ab = 48$, listing the possible factor pairs is an efficient way to find potential solutions.