# Simple algebra question, solve for the variable

#### bobbooey

I'm a little rusty here (or possibly losing my mind at this point), but actually this question involves a bigger problem (substitution rule/calculus) but I'm trying to understand a simple algebra concept first. See the following:

y = 2;
x = 10;

x/y = 5

Ok. I want to solve for y now. Exactly how is this done in algebra? What do you divide or multiply both sides by? I know the answer is y = 1/5x, but I'm unsure as to how one would get to that.

The only way I could see it is to solve for x first:

x = 5y, and then divide 5 on both sides: x/5 = y. There must be an easier and non redundant way, correct?

#### Bacterius

Uhm, you just said it, y = 2 But to solve $$\displaystyle \frac{x}{y} = 5$$, you can rearrange as follows :

>> For $$\displaystyle x$$ :

$$\displaystyle \frac{x}{y} = 5$$

$$\displaystyle \frac{x}{y} \times y = 5 \times y$$

$$\displaystyle \frac{x \times y}{y} = 5 \times y$$

$$\displaystyle \frac{x}{1} = 5 \times y$$ (cancel out the $$\displaystyle y$$)

$$\displaystyle x = 5 \times y$$

$$\displaystyle x = 5y$$

>> For $$\displaystyle y$$ :

$$\displaystyle \frac{x}{y} = 5$$

$$\displaystyle \frac{y}{x} = \frac{1}{5}$$ (inverse on both sides)

$$\displaystyle \frac{y}{x} \times x = \frac{1}{5} \times x$$

$$\displaystyle \frac{xy}{x} = \frac{x}{5}$$

$$\displaystyle \frac{y}{1} = \frac{x}{5}$$ (cancelling out)

$$\displaystyle y = \frac{x}{5}$$

$$\displaystyle y = \frac{1}{5} x$$

This is how I would do it (the long way of course, in real situations I would spare most steps). Does it make sense ?

• dwsmith

#### dwsmith

MHF Hall of Honor
Uhm, you just said it, y = 2 But to solve $$\displaystyle \frac{x}{y} = 5$$, you can rearrange as follows :

>> For $$\displaystyle x$$ :

$$\displaystyle \frac{x}{y} = 5$$

$$\displaystyle \frac{x}{y} \times y = 5 \times y$$

$$\displaystyle \frac{x \times y}{y} = 5 \times y$$

$$\displaystyle \frac{x}{1} = 5 \times y$$ (cancel out the $$\displaystyle y$$)

$$\displaystyle x = 5 \times y$$

$$\displaystyle x = 5y$$

>> For $$\displaystyle y$$ :

$$\displaystyle \frac{x}{y} = 5$$

$$\displaystyle \frac{y}{x} = \frac{1}{5}$$ (inverse on both sides)

$$\displaystyle \frac{y}{x} \times x = \frac{1}{5} \times x$$

$$\displaystyle \frac{xy}{x} = \frac{x}{5}$$

$$\displaystyle \frac{y}{1} = \frac{x}{5}$$ (cancelling out)

$$\displaystyle y = \frac{x}{5}$$

$$\displaystyle y = \frac{1}{5} x$$

This is how I would do it (the long way of course, in real situations I would spare most steps). Does it make sense ?
You went all out.

• Bacterius