Final answer:
The system of equations 8x + 64 = 9y and 3x + 6y = 51 is solved by expressing y in terms of x from the first equation, substituting it into the second, and then solving for x. Subsequently, y is found using the derived x value. The sum of the solutions x and y is 9.
Step-by-step explanation:
To solve the system of equations 8x + 64 = 9y and 3x + 6y = 51, we need to express one variable in terms of the other in one equation, and then substitute it into the other equation. Let's solve the first equation for y:
y = \(\frac{8x + 64}{9}\).
Next, substitute this expression for y into the second equation:
3x + 6\(\frac{8x + 64}{9}\) = 51.
Now, multiply through by 9 to clear the fraction:
27x + 6(8x + 64) = 459.
Simplify and solve for x:
27x + 48x + 384 = 459,
75x + 384 = 459,
75x = 75,
x = 1.
Substitute x back into the equation for y:
y = \(\frac{8(1) + 64}{9}\) = \(\frac{72}{9}\) = 8.
The solutions are x = 1 and y = 8. To find the sum of the solutions, add x and y:
Sum = x + y = 1 + 8 = 9.
The sum of the solutions is 9.