Solve by Substitution x^2+y^2=25 , x+y=8

Problem

{[x2+y2=25],[x+y=8])

Solution

1. Isolate a variable in the linear equation. We can solve the second equation for y

y=8−x

2. Substitute the expression for y into the quadratic equation:

x2+(8−x)2=25

3. Expand the squared binomial:

x2+64−16*x+x2=25

4. Combine like terms and set the equation to zero:

2*x2−16*x+39=0

5. Apply the quadratic formula x=(−b±√(,b2−4*a*c))/(2*a) where a=2 b=−16 and c=39

x=(16±√(,(−16)2−4*(2)*(39)))/(2*(2))

6. Simplify the discriminant:

x=(16±√(,256−312))/4

x=(16±√(,−56))/4

7. Simplify the square root of the negative number using the imaginary unit i

√(,−56)=√(,−1⋅4⋅14)=2*i√(,14)

x=(16±2*i√(,14))/4

x=4±(i√(,14))/2

8. Solve for y by substituting the x values back into y=8−x

y=8−(4±(i√(,14))/2)

y=4∓(i√(,14))/2

Final Answer

(x,y)=(4+(i√(,14))/2,4−(i√(,14))/2),(4−(i√(,14))/2,4+(i√(,14))/2)

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