# Isolating a variable when variable is in both sides of equation

## Homework Statement

y=(2-x)/(5x)

Solving for x=f(y)

I can get as far as 5xy=(2-x)

Tried 5xy=-1(x+2)

All attempts at solving this have given me the wrong answer, and I end up with x on both sides of the equation.

Stuck and not sure what to do.

Correct answer is x=2/(5y+1) , but I don't know how to get there.

Thanks for any help!

symbolipoint
Homework Helper
Gold Member

From the given equation:
1. Multiply both members by 5x.
3. Can you then see to combine two terms of the variable, x ?
4. ...And if you did, then you can finish, finding x as a formula of y.

HallsofIvy
Homework Helper

## Homework Statement

y=(2-x)/(5x)

Solving for x=f(y)

I can get as far as 5xy=(2-x)
This the same as 5xy= 2- x, you don't need the parentheses. If you want x only on the left, then add x to both sides.

Tried 5xy=-1(x+2)

All attempts at solving this have given me the wrong answer, and I end up with x on both sides of the equation.

Stuck and not sure what to do.

Correct answer is x=2/(5y+1) , but I don't know how to get there.

Thanks for any help!

The best way to solve this problem would be to separate the division part, that is to say 2/(5x)- x/(5x) = 2/(5x) - 1/5, then multiply all by 5x to get
5xy=2-x. Now add x to both sides to get 5xy+x = 2. Now do you notice something? We can factor this to x(5y+1)=2, divide by 5y+1 to get x=2/(5y+1)

Bonaparte

The best way to solve this problem would be to separate the division part, that is to say 2/(5x)- x/(5x) = 2/(5x) - 1/5, then multiply all by 5x to get
5xy=2-x. Now add x to both sides to get 5xy+x = 2. Now do you notice something? We can factor this to x(5y+1)=2, divide by 5y+1 to get x=2/(5y+1)

Bonaparte

Aha! Of course...thats what I was missing! I was so close just forgot to factor the x out.

Thanks very much - this problem was really getting under my skin!

Mentallic
Homework Helper
The best way to solve this problem would be to separate the division part, that is to say 2/(5x)- x/(5x) = 2/(5x) - 1/5, then multiply all by 5x to get
5xy=2-x. Now add x to both sides to get 5xy+x = 2. Now do you notice something? We can factor this to x(5y+1)=2, divide by 5y+1 to get x=2/(5y+1)

Bonaparte

Separating the division is both an unnecessary step and makes things harder.

SammyS
Staff Emeritus
Homework Helper
Gold Member

## Homework Statement

y=(2-x)/(5x)

Solving for x=f(y)

An alternate method:
$\displaystyle y=\frac{2-x}{5x}$

$\displaystyle 5y=\frac{2-x}{x}$

$\displaystyle 5y=\frac{2}{x}-1$

$\displaystyle 5y+1=\frac{2}{x}$

$\displaystyle x=\frac{2}{5y+1}$​

symbolipoint
Homework Helper
Gold Member
Separating the division is both an unnecessary step and makes things harder.

NOT necessarily, in case it helps the o.p. to understand.

SammyS
Staff Emeritus
Homework Helper
Gold Member
NOT necessarily, in case it helps the o.p. to understand.
Good point.

In this case, breaking up the rational expression (separating the division) leaves only one term with x in it -- a fairly uncomplicated term at that. This makes it rather easy to solve for x.

Mentallic
Homework Helper
NOT necessarily, in case it helps the o.p. to understand.

I wouldn't think to explain to a student that's working with functions and factorizing that (a+b)/c is the same as a/c+b/c, but I suppose I shouldn't make that assumption.

Mentallic, not that you too seperated the division part, just later.
That was from (2-x)/x to 2/x-x/x. You are right that it is not necessary, but when he gets to much more complicated problems, I personally find it helpful to seperate. Where you seperated you could have multiplied by x both sides and continue, you are perfectly right that it is not "necessary".

Bonaparte

symbolipoint
Homework Helper