Question

Solve the following system of equations by substitution. Show all steps.

f (x) = -x^{2} + 2x +3
g (x) = -2x + 3

Answer 1

Given system of equations:

`\begin{cases}f(x)=-x^2+2x+3\\g(x)=-2x+3\end{cases}`

To solve by substitution, equate the equations and solve for x:

`\begin{aligned}f(x) & = g(x)\\\implies -x^2+2x+3 & = -2x+3\\-x^2+4x & = 0\\x^2-4x & = 0\\x(x-4) & = 0\\\implies x & = 0\\\implies x-4 & = 0 \implies x=4\end{aligned}`

Therefore, the x-values of the solution are
`x = 0` and
`x = 4`.

To find the y-values of the solution, substitute the found values of x into the functions:

`f(0)=-(0)^2+2(0)+3=3`

`g(0)=-2(0)+3=3`

`f(4)=-(4)^2+2(4)+3=-5`

`g(4)=-2(4)+3=-5`

Therefore, the solutions to the given system of equations are:

`(0, 3)` and
`(4, -5)`