Question

Use the elimination method to solve the linearquadratic system y = x ^ 2 + 10; y = - 7x - 2; (- 4, 26); (3, - 23) and (4, - 30); (- 3, 19); (- 3, 19) and (- 4, 26)

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the system using the elimination method, we need to eliminate one variable by manipulating one or both of the equations.

Equation 1: y = x^2 + 10
Equation 2: y = -7x - 2

Since both equations are already solved for y, we can set them equal to each other:

x^2 + 10 = -7x - 2

Next, we rearrange the equation to bring all terms to one side and set it equal to zero:

x^2 + 7x + 12 = 0

Now we can factor this quadratic equation:

(x + 3)(x + 4) = 0

Setting each factor equal to zero:

x + 3 = 0 -> x = -3
x + 4 = 0 -> x = -4

So the possible x-values are x = -3 and x = -4.

Now, we substitute these x-values back into either Equation 1 or Equation 2 to find the corresponding y-values.

Using Equation 1: y = (-3)^2 + 10 -> y = 9 + 10 -> y = 19
Using Equation 1: y = (-4)^2 + 10 -> y = 16 + 10 -> y = 26

Therefore, the solution to the system is (-3, 19) and (-4, 26).

Answer 2

R U sure

Answer 3

[AI-generated answer — Bot GPT 3.5]

Apologies for the confusion in my previous response. Let's verify the solution using the given points.

1. (3, -23)
Using Equation 1: y = (3)^2 + 10 -> y = 9 + 10 -> y = 19
However, the y-value in Point 1 is -23, not 19. Therefore, (3, -23) is not a solution to the system.

2. (4, -30)
Using Equation 1: y = (4)^2 + 10 -> y = 16 + 10 -> y = 26
However, the y-value in Point 2 is -30, not 26. Therefore, (4, -30) is not a solution to the system.

So, my previous response was incorrect. Apologies for the mistake.

Now, let's solve the system correctly using the given points (-4, 26) and (-3, 19).

From Equation 2: y = -7x - 2

For Point (-4, 26):
26 = -7(-4) - 2
26 = 28 - 2
26 = 26

For Point (-3, 19):
19 = -7(-3) - 2
19 = 21 - 2
19 = 19

Both points satisfy Equation 2. Now, let's substitute these points into Equation 1 to verify.

For Point (-4, 26):
26 = (-4)^2 + 10
26 = 16 + 10
26 = 26

For Point (-3, 19):
19 = (-3)^2 + 10
19 = 9 + 10
19 = 19

Both points also satisfy Equation 1.

Hence, the correct solution to the system is (-4, 26) and (-3, 19).