Question

Solve the system of equations:

y= 2x - 2
y= x2 - x-6
O A. (-1,-5) and (4,2)
O B. (0, -2) and (2, 2)
O C. (-1,-4) and (4, 6)
D. (-2,0) and (3,0)

Answer 1

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

Explanation:

`\textsf{Equation 1}:y=2x-2`

`\textsf{Equation 2}:y=x^2-x-6`

Substitute Equation 1 into Equation 2 and solve for x:

`\implies 2x-2=x^2-x-6`

`\implies x^2-3x-4=0`

Find two numbers that multiply to -4 and sum to -3: -4 and 1

Rewrite the middle term as the sum of these two numbers:

`\implies x^2-4x+x-4=0`

Factorize the first two terms and the last two terms separately:

`\implies x(x-4)+1(x-4)=0`

Factor out the common term
`(x-4)`:

`\implies (x+1)(x-4)=0`

`\implies (x+1)=0 \implies x=-1`

`\implies (x-4)=0 \implies x=4`

Substitute the found values of x into Equation 1 and solve for y:

`x=-1 \implies y=2(-1)-2=-4`

`x=4 \implies y=2(4)-2=6`

Therefore, the solution to the system of equations is:

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