[Algebra] The Escalator Trick -- Part 2

Yesterday, I gave some easy examples on "the escalator trick" Part 1 which is a useful short-cut. Today we tackle some slightly more challenging examples. What if the unknown ('x') appears in the denominator? For instance ...

The standard method is to multiply both sides of the equation by 5x, which is able to be cancelled by both 5 and x. After cancelling, we have 2x on the RHS. To get rid of the 2, we divide both sides of the equation by 2. After the cancellation, we get 10 = x. By the Law of Symmetry (if A=B, then B=A), we conclude that x = 10. I am sure most of you will understand this. But this equation takes a whopping 6 steps to unravel. Is there a faster way?

We can use the escalator trick introduced yesterday. You know those escalators going up and down you see in shopping malls (if you still visit them nowadays)? Imagine escalators going between the top-right and the bottom left, and escalators going between the top-left and the bottom right. They criss-cross each other.

Now, i want x to be on top (as numerator) so I move it from the lower-left to the top-right. But x has to be alone, so the 2 and the 5 must move. 2 goes from top-right to lower-left and 5 goes from lower-right to upper-left. These are indicated by the red arrows. They move all at once! After moving, you cancel and simplify. Do not multiply 4 × 5. It is better to cancel first, so that you get smaller numbers, which are easier to manage. We get 10 = x, but we can use the Law of Symmetry to write it as x = 10.

Here is another example.

Move the 'x' from the lower right to the upper left. 'x' wants to be left alone (no pun intended), so 9 and 5 have to go. '9' takes the escalator down to the lower-right and '5' takes the escalator up to the upper-right.

 Do not multiply -- cancel first and simplify.

We get 10/3. If you know the escalator trick, the equation can be solved in three steps!

Of course you should understand the orthodox standard methods first. After all, the escalator trick is based on the principles of multiplying or dividing the same quantities on both sides of the equation. Once you get past that, then the escalator trick can help you quickly solve certain types of equations! Algebra is actually very easy!