Question

Which equation is not a linear function? Select each correct function.

y = xy + 2

y = x + 2y

y = -x - 2y

y = x - y + 2

y = x^2 - 2 + 2y

y = -y + 2 - 2x

Answer 1

$\text{The point-slope form of a linear function:}$

$y = mx + b$

$y=xy+2\qquad\text{subtract xy from both sides}\\\\y-xy=2\qquad\text{distributive}\\\\(1-x)y=2\qquad\text{divide both sides by (1 - x)}\\\\y=\dfrac{2}{1-x}\to\text{nonlinear}\\---------------------------\\y=x+2y\qquad\text{subtract 2y from both sides}\\\\-y=x\qquad\text{change the signs}\\\\y=-x\to\text{linear}\\-----------------------------\\y=-x-2y\qquad\text{add 2y to both sides}\\\\3y=-x\qquad\text{divide both sides by 3}\\\\y=-\dfrac{1}{3}x\to\text{linear}\\-----------------------------$

$y=x-y+2\qquad\text{add y to both sides}\\\\2y=x+2\qquad\text{divide both sides by 2}\\\\y=\dfrac{1}{2}x+1\to\text{linear}\\----------------------------\\y=x^2-2+2y\qquad\text{subtract 2y from both sides}\\\\-y=x^2-2\qquad\text{change the signs}\\\\y=-x^2+2\to\text{nonlinear}\\----------------------------\\y=-y+2-2x\qquad\text{add y to both sides}\\\\2y=2-2x\qquad\text{divide both sides by 2}\\\\y=-x+1\to\text{linear}$