Which of the following uses the substitution method to solve the linear-quadratic system y = x^2 - x - 40, y = -3x - 5?
A. (-7, 26)
B. (5, -10)
C. (5, -10) and (-7, 26)
D. (-5, 20) and (7, -16)
To solve the linear-quadratic system using the substitution method, we substitute the value of y from one equation into the other equation. Let's substitute y = -3x - 5 into the equation y = x^2 - x - 40:
-3x - 5 = x^2 - x - 40
Rearranging the equation, we get:
x^2 - x + 3x - 5 + 40 = 0
x^2 + 2x + 35 = 0
This quadratic equation does not factor, so we can use the quadratic formula to find the solutions for x:
x = (-2 ± √(2^2 - 4(1)(35))) / (2(1))
x = (-2 ± √(4 - 140)) / 2
x = (-2 ± √(-136)) / 2
x = (-2 ± 2√(-34)i) / 2
x = -1 ± √(-34)i
Since we are looking for real solutions, there are no real values of x that satisfy the equation. Therefore, there are no real solutions to the linear-quadratic system.
The answer is none of the above.