Question

HELP ! Solve the following system of equations and show all work. y = −x^2+4 y = 2x+1

Answer 1

$\begin{cases} y=-x^{2}+4 \\ y=2x+1 \end{cases} \\ \\ 2x+1=-x^{2}+4 \\ \\ x^{2}+2x-3=0 \\ \\ x^{2}+3x-x-3=0 \\ \\ x(x+3)-(x+3)=0 \\ \\ (x-1)(x+3)=0 \\ \\ x-1=0 \ \vee \ x+3=0 \\ \\ x=1 \imlpies y=2*1+1=3 \\ \\ x=-3 \implies y=2*(-3)+1=-5 \\ \\ \boxed{x=1 \ \wedge \ y=3 \ \vee \ x=-3 \ \wedge \ y=-5}$

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Answer 2

Since both equations have y on the left-hand side, set them equal:

2x+1=-x^2+4

Add x^2 and subtract 4:

x^2+2x-3=0

Factor to get:

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

Thus, x=1 or -3.  Substituting in for x, y=2*1+1=3 or y=-3*2+1=-5.  Therefore, our ordered pairs are (1, 3) and (-3, -5).