Solve the system of equations by substitution.

Question

y=x^2-4x-6
y=-2x^2-5x-2

Answer 1

To solve the system by substitution, we can set the expressions for y equal to each other because they both equal y:

x^2 - 4x - 6 = -2x^2 - 5x - 2

Next, we will combine like terms and solve for x. Let's bring all the terms to one side:

x^2 + 2x^2 - 4x + 5x - 6 + 2 = 0

Combine like terms:

3x^2 + x - 4 = 0

Now, we have a quadratic equation. Let's use the quadratic formula to solve for x where a = 3, b = 1, and c = -4:

x = [-b ± √(b^2 - 4ac)] / (2a)

x = [-1 ± √(1^2 - 4(3)(-4))] / (2 * 3)

x = [-1 ± √(1 + 48)] / 6

x = [-1 ± √(49)] / 6

x = [-1 ± 7] / 6

We now have two possible solutions for x:

x = (-1 + 7) / 6
x = 6 / 6
x = 1

And

x = (-1 - 7) / 6
x = -8 / 6
x = -4/3 (or -1.333... when written as a decimal)

Now substitute these x values back into either original equation to solve for y. Let's use the first equation:

For x = 1:

y = (1)^2 - 4(1) - 6
y = 1 - 4 - 6
y = -9

So, one solution is (x, y) = (1, -9).

For x = -4/3:

y = (-4/3)^2 - 4(-4/3) - 6
y = 16/9 + 16/3 - 6
y = 16/9 + 16/3 - 6 * 3/3
y = 16/9 + 48/9 - 18/9
y = (16 + 48 - 18) / 9
y = 46/9

So, another solution is (x, y) = (-4/3, 46/9).

The complete solution set to this system of equations by substitution is:
(x, y) = (1, -9) and (x, y) = (-4/3, 46/9)