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Solve Pairs of Equations Where One Equation is Quadratic

Solve the system of equations:

$x^{2}-5x+1=y$
$x+y+2=0$

Solve 2. for y.

$x+y+2=0$

$x+y+2-2=0-2$

$x+y=-2$

$x+y-x=-2-x$

$y=-2-x$

Plug into 1.

$x^{2}-5x+1=-2-x$

$x^{2}-5x+1+2=-2-x+2$

$x^{2}-5x+3=-x$

$x^{2}-5x+3+x=-x+x$

$x^{2}-4x+3=0$

$(x+$ ___ $)(x+$ ___ $)$

Factors of 3: ±3, ±1

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

$x-3=0\rightarrow x-3+3=0+3\rightarrow x=3$

$x-1=0\rightarrow x-1+1=0+1\rightarrow x=1$

Plug in x and solve for y in 2.

$x+y+2=0$

$3+y+2=0\rightarrow y+5=0\rightarrow y+5-5=0-5\rightarrow y=-5$

$x=3,\ y=-5$

$1+y+2=0\rightarrow y+3=0\rightarrow y+3-3=0-3\rightarrow y=-3$

$x=1,\ y=-3$