Elementary Algebra

a) We first simplify by letting $v=1/x$ and $u = 1/y$. Then, we multiply by the necessary value to make a variable cancel and solve. Finally, we plug the value of the variable we found in order to find the missing variable. $v + 2u = 7/12 \\ 3v - 2u = 5/12$ Thus: $4v = 1 \\ v = .25$ And: $u = 1/6$ Thus, x=4 and y=6. b) We first simplify by letting $v=1/x$ and $u = 1/y$. Then, we multiply by the necessary value to make a variable cancel and solve. Finally, we plug the value of the variable we found in order to find the missing variable. $2v + 3u = 19/15 \\ -2v + u = -7/15$ Thus: $4u = 12/15 \\u = 1/5$ And: $v = 1/3$ Thus, x=3 and y=5. c) We first simplify by letting $v=1/x$ and $u = 1/y$. Then, we multiply by the necessary value to make a variable cancel and solve. Finally, we plug the value of the variable we found in order to find the missing variable. $3v - 2u = 13/6 \\ 2v + 3u = 0$ Thus: $-13u = 13/3 \\ u =-1/3$ And: $v = 1/2$ Thus, x=2 and y=-3. d) We first simplify by letting $v=1/x$ and $u = 1/y$. Then, we multiply by the necessary value to make a variable cancel and solve. Finally, we plug the value of the variable we found in order to find the missing variable. $4v + u = 11 \\ 3v -5u = -9$ Thus: $23v = 46 \\ v = 2$ And: $u = 3$ Thus, x=1/2 and y=1/3. e) We first simplify by letting $v=1/x$ and $u = 1/y$. Then, we multiply by the necessary value to make a variable cancel and solve. Finally, we plug the value of the variable we found in order to find the missing variable. $5v - 2u = 23 \\ 4v + 3u = 23/2$ Thus: $23v = 92 \\ v = 4$ And: $u = -3/2$ Thus, x=1/4 and y=-2/3. f) We first simplify by letting $v=1/x$ and $u = 1/y$. Then, we multiply by the necessary value to make a variable cancel and solve. Finally, we plug the value of the variable we found in order to find the missing variable. $2v - 7u = 9/10 \\ 5v + 4u = -41/20$ Thus: $-43u = 8.6 \\ u = -1/5$ And: $v = -1/4$ Thus, x=-4 and y=-5.