Question

What are the solutions to the following system of equations?

y = x² + 12x + 30
8x - y = 10

A. (-4,-2) and (2, 5)
B. (-2,-4) and (2,5)
C. (-2,-4) and (5,2)
D. No real solutions

Answer 1

D. no real answer

Explanation:

8x-y=10 then y=8x-10

`y = {x}^(2) + 12x + 30 = 8x - 10`

`{x}^(2) + 4x + 40 = 0`

it's a grade 2 equation, it has no real answer

because

`{4}^(2) - 4(1)(40) = - 144 < 0`

Answer 2

D. No real solutions

Explanation:

Any of the usual methods of finding solutions to the system of equations will show there are none,

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graphing

The graphs of the two equations do not intersect. This means any of the answer choices that identifies specific solution points must be incorrect. There are no real solutions.

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check the answers

You can check to see if any of the proposed solution points satisfy the equations.

A: (x, y) = (-4, -2) ⇒ 8(-4) -(-2) = -30 ≠ 10 . . . not a solution

B: (x, y) = (-2, -4) ⇒ 8(-2) -(-4) = -12 ≠ 10 . . . not a solution

C: same (x, y) as B.

So, none of these answer choices describes a solution to the system of equations. We conclude there are ...

No real solutions

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solve the system

Using the first equation, you can substitute for y in the second equation:

8x -(x² +12x +30) = 10

x² +4x +40 = 0 . . . . . . . put in standard form

(x +2)² +36 = 0 . . . . . . . write in vertex form

There are no real values of x that will make (x+2)² be negative, so there are no real solutions.