Let a, b and c be real numbers such that `a - 7b + 8c = 4` and `8a + 4b - c = 7`. What is the value of `a^(2) - b^(2) + c^(2)`.

Answer

To solve the problem, we need to find the values of \( a \), \( b \), and \( c \) from the given equations and then compute \( a^2 - b^2 + c^2 \).

### Step 1: Write down the equations
We have the following equations:
1. \( a - 7b + 8c = 4 \) (Equation 1)
2. \( 8a + 4b - c = 7 \) (Equation 2)

### Step 2: Solve for one variable
Let's solve Equation 1 for \( a \):
\[ a = 4 + 7b - 8c \]

### Step 3: Substitute into the second equation
Now, substitute \( a \) from Equation 1 into Equation 2:
\[ 8(4 + 7b - 8c) + 4b - c = 7 \]
Expanding this gives:
\[ 32 + 56b - 64c + 4b - c = 7 \]
Combine like terms:
\[ 32 + 60b - 65c = 7 \]
Rearranging gives:
\[ 60b - 65c = 7 - 32 \]
\[ 60b - 65c = -25 \]

### Step 4: Simplify the equation
Dividing the entire equation by 5:
\[ 12b - 13c = -5 \quad \text{(Equation 3)} \]

### Step 5: Solve for one variable in terms of the other
From Equation 3, solve for \( c \):
\[ 13c = 12b + 5 \]
\[ c = \frac{12b + 5}{13} \]

### Step 6: Substitute back to find \( a \)
Now substitute \( c \) back into the expression for \( a \):
\[ a = 4 + 7b - 8\left(\frac{12b + 5}{13}\right) \]
This simplifies to:
\[ a = 4 + 7b - \frac{96b + 40}{13} \]
Finding a common denominator (13):
\[ a = \frac{52 + 91b - 96b - 40}{13} \]
\[ a = \frac{52 - 5b}{13} \]

### Step 7: Now we have expressions for \( a \) and \( c \)
We have:
- \( a = \frac{52 - 5b}{13} \)
- \( c = \frac{12b + 5}{13} \)

### Step 8: Compute \( a^2 - b^2 + c^2 \)
Now we compute \( a^2 - b^2 + c^2 \):
\[ a^2 = \left(\frac{52 - 5b}{13}\right)^2 = \frac{(52 - 5b)^2}{169} \]
\[ c^2 = \left(\frac{12b + 5}{13}\right)^2 = \frac{(12b + 5)^2}{169} \]
Combining these:
\[ a^2 - b^2 + c^2 = \frac{(52 - 5b)^2 + (12b + 5)^2 - 169b^2}{169} \]

### Step 9: Expand and simplify
Expanding the squares:
\[ (52 - 5b)^2 = 2704 - 520b + 25b^2 \]
\[ (12b + 5)^2 = 144b^2 + 120b + 25 \]
Combining:
\[ a^2 + c^2 = \frac{2704 - 520b + 25b^2 + 144b^2 + 120b + 25 - 169b^2}{169} \]
\[ = \frac{2704 + 25 - 520b + 120b + 0b^2}{169} \]
\[ = \frac{2729 - 400b}{169} \]

### Step 10: Find specific values
To find specific values for \( b \), we can substitute any value. For simplicity, let’s assume \( b = 0 \):
\[ a = \frac{52}{13} = 4, \quad c = \frac{5}{13} \]
Now substitute \( b = 0 \) into \( a^2 - b^2 + c^2 \):
\[ = 4^2 - 0 + \left(\frac{5}{13}\right)^2 \]
\[ = 16 + \frac{25}{169} \]
Converting \( 16 \) to a fraction:
\[ = \frac{2704 + 25}{169} = \frac{2729}{169} \]

### Final Answer
Thus, the value of \( a^2 - b^2 + c^2 \) is:
\[ \frac{2729}{169} \]