Question

Use the elimination method to solve the linear-quadratic system y = x^2 + x -42, y =- x+6.

• (6,0)
• (-8,14)
• (8, -2) and (-6,12)
• (-8, 14) and (6, 0)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the linear-quadratic system using the elimination method, we need to eliminate one of the variables so that we can solve for the other variable.

1) In the given system of equations:
y = x^2 + x - 42 ...(Equation 1)
y = -x + 6 ...(Equation 2)

2) We can start by eliminating the variable "y" from the system. We can do this by equating the right sides of both equations:
x^2 + x - 42 = -x + 6

3) Rearranging the equation:
x^2 + 2x - 48 = 0

4) Factoring the equation:
(x + 8)(x - 6) = 0

5) Setting each factor equal to zero:
x + 8 = 0 or x - 6 = 0

6) Solving for "x":
If x + 8 = 0, then x = -8
If x - 6 = 0, then x = 6

So, the possible solution for the system is either (-8, 14) or (6, 0).

Checking each solution:
For (-8, 14):
From Equation 2: y = -(-8) + 6 = 8 + 6 = 14
So, the solution (-8, 14) satisfies both equations.

For (6, 0):
From Equation 2: y = -6 + 6 = 0
However, from Equation 1: y = 6^2 + 6 - 42 = 36 + 6 - 42 = 42 - 42 = 0
So, the solution (6, 0) satisfies both equations.

Therefore, the solution is (6, 0) and (-8, 14).