Solve 1/x + 1/y = 8 for y in terms of x.

Step-by-step solution

1. 1

   Note that one equation in two unknowns has infinitely many solutions. The task is therefore not to find numbers but to express $y$ as a function of $x$ — a curve of solutions rather than a point. The domain excludes $x=0$ and $y=0$.

2. 2

   Combine the left side over a common denominator.

   $\frac{1}{x}+\frac{1}{y}=\frac{y+x}{xy}=8$

3. 3

   Clear the denominator. Multiply both sides by $xy$ (nonzero by the domain):

   $x+y=8xy$

4. 4

   Collect the $y$ terms and factor. Move $y$ to the right and factor it out:

   $x=8xy-y=y(8x-1)$

5. 5

   Divide and state the exclusions. Dividing by $8x-1$ needs $8x-1\neq 0$:

   $y=\frac{x}{8x-1},\qquad x\neq 0,\ x\neq\frac{1}{8}$

   The value $x=\tfrac18$ is excluded because then $\tfrac1x=8$ already, leaving $\tfrac1y=0$, which is impossible.

6. 6

   Check two points on the curve. At $x=1$: $y=\tfrac17$, and $1+7=8$ $\checkmark$. At $x=\tfrac14$: $y=\tfrac{1/4}{1}=\tfrac14$, and $4+4=8$ $\checkmark$.