Question

Solve this nonlinear system of equations.

y=-x^2+6x-5
y=3
step 1: use substitution to combine the equations. Rewrite so that one side is equal to zero.
step 2: factor the equation
step 3: identify the x-values of the solutions.
step 4: identify the solutions to the system.

Answer 1

1.
`x^2-6x+8=0`

2.
`(x-4)(x-2)=0`

3.
`x=4 \text{ and } x=2`

4.
`x=4 \text{ and } x=2`

Explanation:

1. When using substitution all we do would be is substituse the y for a 3.

This leaves us with the equation:

`3=-x^2+6x-5`

Rewriting it we get:

`0=-x^2+6x-8`

or if we shift the 0 to the other side:

`x^2-6x+8=0`

2. In order to factor the equation we can use the butterfly method:

`\left[\begin{array}{ccc}1&-4\\1&-2\end{array}\right]`
So it factors out to:

`(x-4)(x-2)=0`
You can also use the quadratic formula.

3. To find the solutions we just set each factor to 0

`x-4=0\\\text{and}\\x-2=0`
So the x-values would be:

`x=4 \text{ and } x=2`

4. To find the solution to the system we just plug in the values and it turns out to be the same numbers as before.