Written by tutor Steve C.
Algebra is all about solving for X (and Y, and Z, but we start with only X).
Pre-algebra is about learning manipulations so we can find and isolate X.
Cross multiplication is a way of manipulating terms to get X here and everything else there.
That can basically be called “Isolating X”; putting it on one side, so only numbers are on the other side.
There are several “next steps” after cross multiplication, some more obvious than others. The “downstairs swap” is simply a tool that replaces cross multiplication. It eliminates the mechanics of that cross step, as well as several subsequent steps. It can be derived and/or proven with cross multiplication, but that’s not what you need here.
There are several ways to solve a problem. All of them involve isolating X. There are several ways to manipulate numbers so that you can isolate X. Cross Multiplication is one way to do so, when you have 4 terms to deal with. The Down Stairs Swap is one way to do so, when you have 3 terms to deal with.
The bottom line is: get X on one side of the equation, and get everything else on the other side.
Down Stairs Swap
Let’s review a well known concept...
And its less well known cousin...
DOWN STAIRS SWAP
We all know about Cross Multiplication:
When A/B = C/D, then AD = BC
But what happens when D = 1 ?
Then you get A/B = C
Multiply by B
and you get A = CB
Divide by C
and you get A/C = B
Let’s consider these two equations:
C = A/B B = A/C
Whenever you have a solo term on one side, and
terms upstairs and downstairs on the other side,
then you can swap the solo term with the downstairs.
That’s the Downstairs Swap.
(Of course, this presumes that nothing downstairs is zero. However, if it is, then the initial Cross Multiplication won’t work, either.)
We can find several uses for this simple trick. Learn how to apply it to unit rates.