Systems of Equations (Elimination) Practice AK
1) $\require{cancel}x=\dfrac{9}{6}$ or $\dfrac{3}{2}$
$$\eqalign{2x+\cancel{3y}&=6\\4x\cancel{-3y}&=3\\\hline 6x\qquad&=9\\x&=\dfrac{9}{6}}$$
2) $y=1$
Plug the x value above into an equation from the system:
$$\eqalign{2(\dfrac{9}{6})+3y&=6\\\dfrac{18}{6}+3y&=6\\3+3y&=6\\3y&=3\\y&=1}$$
3) $x=32$
$$\eqalign{\cancel{x}+2y&=8\\\cancel{-x}-3y&=4\\\hline-y&=12\\y&=-12}$$
Plug the y value into an equation from the system to find the x value:
$$\eqalign{x+2(-12)&=8\\x-24&=8\\x=32}$$
4) $y=-12$
5) $x=\dfrac{-2}{3}$
$$\eqalign{5x-4y&=2\\-x-y&=2\\\text{Multiply second equation by 5}\\\cancel{5x}-4y&=2\\\cancel{-5x}-5y&=10\\\hline-9y&=12\\y&=\dfrac{12}{-9}=\dfrac{-4}{3}\\\text{Plug in y value to solve for x}\\-x-(\dfrac{-4}{3})&=2\\-x+\dfrac{4}{3}&=2\\-x&=\dfrac{2}{3}\\x&=\dfrac{-2}{3}}$$
6) $y=\dfrac{-4}{3}$.
7) $(\dfrac{-1}{5},1\dfrac{2}{5})$
$$\eqalign{y&=2+3x\\-2y&=4x-2\\\text{Rearrange equations so that like terms line up}\\y&=2+3x\\-2y&=-2+4x\\\text{Multiply equation one by 2}\\\cancel{2y}&=4+6x\\\cancel{-2y}&=-2+4x\\\hline 0&=2+10x\\-2&\quad-2\\-2&=10x\\\dfrac{-2}{10}&=x\\\dfrac{-1}{5}&=x\\\text{Plug x into an equation to find y}\\y&=2+3(\dfrac{-1}{5})\\y&=2+\dfrac{-3}{5}\\y=1\dfrac{2}{5}\\\text{Solution=}(\dfrac{-1}{5},1\dfrac{2}{5})}$$
8) $(0,2)$
$$\eqalign{3x-y&=-2\\-x&=6y-12\\\text{Rearrange equations so like terms align}\\3x-y&=-2\\-x-6y&=-12\\\text{Multiply second equation by 3}\\\cancel{3x}-y&=-2\\\cancel{-3x}-18y=-36\\\hline-19y&=-38\\y&=2\\\text{Plug y into equation to solve for x}\\3x-(2)&=-2\\3x&=0\\x&=0}$$
9) No solution.
$$\eqalign{4x-4y&=4\\y&=x+2\\\text{Rearrange equations to line up like terms}\\4x-4y&=4\\-x+y&=2\\\text{Multiply second equation by 4}\\4x-4y&=4\\-4x+4y=8\\\hline0&=12\\\text{Not true; There is NO SOLUTION (these lines are parallel).}}$$
10) $(\dfrac{-1}{13},\dfrac{-8}{13})$
$$\eqalign{-2y&=1-3x\\-2x&=3y+2\\\text{Rearrange equations to line up like terms}\\-2y&=1-3x\\-3y&=2+2x\\\text{Multiply the first equation by 2 and the second equation by 3}\\-4y&=2\cancel{-6x}\\-9y&=6+\cancel{6x}\\\hline -13y&=8\\y&=\dfrac{8}{-13}\\\text{Plug y into equation to solve for x}\\-2(\dfrac{8}{-13})&=1-3x\\\dfrac{16}{13}&=1-3x\\-1&=-1\\\dfrac{3}{13}&=-3x\\\dfrac{-1}{13}&=x\\\text{Solution:}(\dfrac{-1}{13},\dfrac{-8}{13})}$$