Question

Use the elimination method to solve the linear-quadratic system y=x2−2x−2, y=2x−6.(1 point)
Responses

(2,−2) and (−2,2)
left parenthesis 2 comma negative 2 right parenthesis and left parenthesis negative 2 comma 2 right parenthesis

(−2,−10)
left parenthesis negative 2 comma negative 10 right parenthesis

(2,−2)
left parenthesis 2 comma negative 2 right parenthesis

(−2,−2) and (−2,2)
left parenthesis negative 2 comma negative 2 right parenthesis and left parenthesis negative 2 comma 2 right parenthesis

Answer 1

To solve the system of equations using the elimination method, we need to eliminate one of the variables. In this case, we can eliminate the variable y by multiplying the second equation by -1.

Original equations:
y = x^2 - 2x - 2
y = 2x - 6

Multiply the second equation by -1:
-y = -2x + 6

Now add the two equations together:
(y) + (-y) = (x^2 - 2x - 2) + (-2x + 6)

This simplifies to:
0 = x^2 - 4x + 4

To solve this quadratic equation, we can factor it:
0 = (x - 2)(x - 2)

The factored form tells us that x = 2 is the only solution for x.

Now substitute this value back into one of the original equations to solve for y:
y = 2(2) - 6
y = -2

Therefore, the solution to the system of equations is (2, -2).