Solve the following system of equations.

Question

y=x^2 -6 and 8x+y=-18

Thank you for answering this

Answer 1

y = x² - 6 is Equation 1

8x + y = -18 is Equation 2

And we want to know where the line crosses the parabola

Since we already have one equation as

y = x² - 6

We can easily try a substitution.

Substitute x² - 6 into Equation 2 and solve for x

8x + x² - 6 = -18

Add 18 to both sides of the equation

8x + x² - 6 + 18 = 0

Combine like terms

8x + x² + 12 = 0

We have a quadratic

x² + 8x + 12 =0

This can be factored

(x + 6)(x +2)=0

x = -6

x = -2

Now we can substitute these values back into y = x² - 6 and solve for y

y = (-6)² - 6 = 36 - 6 = 30 and we have (-6, 30)

For the next value of x

y = x² - 6

y = (-2)² - 6 = 4 - 6 = -2 we have (-2, -2)

We can check both these values by plugging back into Equation 2

8x + y = -18

8(-6) + 30 = -18

-48 + 30 = -18

-18 = -18

For the next pair

8x + y = -18

8(-2) + (-2) = -18

(-16) + (- 2) = -18

-16 - 2 = -18

-18 = -18

So the line crosses the parabola at two points (-6, 30) and (-2, -2)

You might want to check that on graphing calculator or computer

I hope you find this useful and if you have any questions please send me an email.

Answer 2

Hello!

You are trying to figure out where these two equations intersect. You could graph both of them and visually see the points where they cross OR you could use substitution.

1. Graphing .... check out DESMOS.com, a free online graphing calculator! It's awesome. Make sure each equation is solved for y. Then type each equation into Desmos and you will quickly see where they intersect.
2. Substitution... solve the second equation for y to get y = -8x - 18x. Then plug this into the first equation everywhere you see the variable y.

y = x² -6

-8x - 18 = x² - 6 (add 8x and 18 to both sides)

0 = x² +8x + 12 (factor the polynomial)

0 = (x + 2) (x + 6)

x+2 = 0 and x+6 = 0

x = -2 and x = -6

Now you have to find the y values that correspond with these x values.

I would use the original linear equation: y = -8x - 18

When x = -2,

y = -8(-2) - 18

y = 16 - 18

y = -2

The solution is (-2, -2).

When x = -6,

y = -8(-6) - 18

y = 48 - 18

y = 30

The solution is (-6, 30).

Our two solutions, or points of intersection, are (-2, -2) and (-6, 30).

Let me know if you need clarification on anything. Have a great day!