Question

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?

Answer 1

see explanation

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 )

Answer 2

The answer to your question is below

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)