Use the elimination method to solve the linear-quadratic system y=x^2+2x−4 , y=−x−6 .(1 point) The solutions are (−1,−5) and ().
To solve the system of equations using the elimination method, we need to eliminate one variable by adding or subtracting the equations. In this case, we can eliminate the variable y by adding the two equations together.
We have:
y = x^2 + 2x - 4
y = -x - 6
Adding the equations:
(x^2 + 2x - 4) + (-x - 6) = 0
Simplifying:
x^2 + x - 10 = 0
Now we can solve this quadratic equation by factoring or using the quadratic formula. Factoring gives us:
(x + 5)(x - 2) = 0
Setting each factor equal to zero:
x + 5 = 0 or x - 2 = 0
Solving for x:
x = -5 or x = 2
Now substitute these values of x back into one of the original equations to find the corresponding values of y.
For x = -5:
y = (-5)^2 + 2(-5) - 4
y = 25 - 10 - 4
y = 11
So one solution is (-5, 11).
For x = 2:
y = (2)^2 + 2(2) - 4
y = 4 + 4 - 4
y = 4
So the other solution is (2, 4).
Therefore, the solutions to the linear-quadratic system are (-5, 11) and (2, 4).