A parabola has the equation y=x^2-x-6. A line has a slope 2 and y-intercept - 2. Think of the system of equations that you can create with this information. What is a good strategy to use in order to solve the system or equations?

see explanation

Step-by-step explanation:

The equation of a line in slope - intercept form is

y = mx + c (m is the slope and c the y - intercept)

Here m = 2 and c = - 2, thus

y = 2x - 2 ← equation of line

Hence system of equations is

y = x² - x - 6 → (1)

y = 2x - 2 → (2)

Since both equations express y in terms of x we can equate the right sides

x² - x - 6 = 2x - 2 (subtract 2x - 2 from both sides)

x² - 3x - 4 = 0 ← in standard form

(x - 4) (x + 1) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 4 = 0 ⇒ x = 4

x + 1 = 0 ⇒ x = - 1

Substitute these values into (2) for corresponding values of y

x = 4 : y = (2 * 4) - 2 = 8 - 2 = 6 ⇒ (4, 6)

x = - 1 : y = (2 * - 1) - 2 = - 2 - 2 = - 4 ⇒ ( - 1, - 4)

Solutions to the system of equations are

( - 1, - 4) and (4, 6)

The answer to your question is below

Step-by-step explanation:

Data

Parabola y = x² - x - 6

Line slope = 2 and y - intercept = - 2

Process,

Write the equation of the line,

y = mx + b

m = 2

b = - 2

y = 2x - 2

To solve this system of equations equal both "y" to have only one equation in terms of "x".

2x - 2 = x² - x - 6

Equal to zero

x² - x - 6 - 2x + 2 = 0 and simplify

x² - 3x - 4 = 0

Solve the quadratic equation by factoring

(x - 4) (x + 1) = 0

Find the x values

x₁ = 4 and x₂ = - 1

Finally find the y values

y₁ = 2 (4) - 2 = 8 - 2 = 6

y₂ = 2 (-1) - 2 = - 2 - 2 = - 4

Give the result

P₁ (4, 6) P₂ (-1, - 4)