Rearranging Equations – Equations with Common Terms on Both Sides of the Equals Sign

When faced with a problem like this, we can turn to our trusted equation rearranging weapons, inverse operations and factorisation.

At first glance, rearranging this equation below from fluid mechanics:

rho g h subscript A space plus space 1 half space rho u subscript A squared space plus space P subscript A space equals space rho g h subscript B space plus space 1 half space rho u subscript B squared space plus space P subscript B

To isolate left parenthesis P subscript A space minus space P subscript B right parenthesis and to get to this:

left parenthesis P subscript A space minus space P subscript B right parenthesis space equals space rho g space left parenthesis h subscript B minus h subscript A right parenthesis space plus space 1 half space rho space left parenthesis space u subscript B squared space minus space u subscript A squared space right parenthesis

It looks like a nightmare, but it doesn’t involve anything we haven’t already done.

Let’s take it a step at a time. First, we subtract  P subscript B from both sides to give us:

rho g h subscript A space plus space 1 half space rho u subscript A squared space plus space P subscript A minus space P subscript B equals space rho g h subscript B space plus space 1 half space rho u subscript B squared

Then subtract rho g h subscript A from both sides to give us:

1 half space rho u subscript A squared space plus space P subscript A minus space P subscript B equals space rho g h subscript B space plus space 1 half space rho u subscript B squared space minus space rho g h subscript A

Then subtract 1 half space rho u subscript A squared from both sides to give us:

P subscript A minus space P subscript B equals space rho g h subscript B space plus space 1 half space rho u subscript B squared space minus space rho g h subscript A space minus space 1 half space rho u subscript A squared

Now we can factorise the right side by rho g:

P subscript A minus space P subscript B equals space rho g space left parenthesis h subscript B space end subscript minus space h subscript A space end subscript right parenthesis space plus space 1 half space rho u subscript B squared space minus space 1 half space rho u subscript A squared

And factorise by 1 half rho:

left parenthesis P subscript A minus space P subscript B right parenthesis equals space rho g space left parenthesis h subscript B space end subscript minus space h subscript A space end subscript right parenthesis space plus space 1 half space rho space open parentheses u subscript B squared space minus space u subscript A squared close parentheses

We have to be careful with the signs when factorising, but if you use the + or – directly before the term, you won’t go wrong.

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