Question

Solve the system of equations by substitution.

y=x^2-5x-5
y=-2x^2+6x-1

Answer 1

[AI-generated answer — Bot GPT-4 Turbo]

To solve the system of equations using the substitution method, we'll set the two equations equal to each other because they both equal y:

x^2 - 5x - 5 = -2x^2 + 6x - 1

Now we'll combine like terms by bringing all terms to one side of the equation:

x^2 + 2x^2 - 5x - 6x - 5 + 1 = 0

3x^2 - 11x - 4 = 0

We have a quadratic equation, which we will attempt to solve for x. This equation can be factored, and if it cannot be factored easily, you could use the quadratic formula.

3x^2 - 12x + x - 4 = 0

Group the terms:

(3x^2 - 12x) + (x - 4) = 0

Factor out the common factors:

3x(x - 4) + 1(x - 4) = 0

Now factor by grouping:

(3x + 1)(x - 4) = 0

Set each factor equal to zero to solve for x:

3x + 1 = 0 or x - 4 = 0

x = -1/3 or x = 4

Now we can use these solutions for x to solve for y using either of the two original equations. Let's use the first equation:

For x = -1/3:
y = (-1/3)^2 - 5(-1/3) - 5
y = 1/9 + 5/3 - 5
y = 1/9 + 15/9 - 45/9
y = 16/9 - 45/9
y = -29/9

For x = 4:
y = (4)^2 - 5(4) - 5
y = 16 - 20 - 5
y = -9

So the solutions to the system of equations are:

(x, y) = (-1/3, -29/9) and (x, y) = (4, -9)