Sure, let's consider each system of equations individually and determine the number of real solutions for each.
System A:
The system consists of:
[tex]\[ x^2 + y^2 = 17 \][/tex]
[tex]\[ y = -\frac{1}{2}x \][/tex]
Through analysis, we find that this system has 2 real solutions.
System B:
The system consists of:
[tex]\[ y = x^2 - 7x + 10 \][/tex]
[tex]\[ y = -6x + 5 \][/tex]
Through analysis, we observe that this system has 2 real solutions.
System C:
The system consists of:
[tex]\[ y = -2x^2 + 9 \][/tex]
[tex]\[ 8x - y = -17 \][/tex]
Through analysis, it is determined that this system has 1 real solution.
So, the final number of solutions for each system is:
System A has \( \boxed{2} \) real solutions.
System B has \( \boxed{2} \) real solutions.
System C has [tex]\( \boxed{1} \)[/tex] real solution.
