Can you divide both sides of an equation by x?
Updated
The short answer
Dividing both sides by x only works when x isn't 0, so it can throw away the solution x = 0. For x² = 5x, move everything to one side and factor: x(x − 5) = 0, so x = 0 or x = 5.
What it looks like
Here is x² = 5x solved two ways.
| Wrong | Right |
|---|---|
| Divide both sides by x: x = 5 | Subtract 5x from both sides: x² − 5x = 0 |
| Answer: x = 5 | Factor: x(x − 5) = 0, so x = 0 or x = 5 |
The wrong answer passes a check, because x = 5 is a real solution, but it's incomplete. Try x = 0 in the original: 0² = 0 and 5 × 0 = 0, so x = 0 works too. Dividing by x threw it away.
In alai, an iPad app that reads your handwriting as you work, the tutor, Ally, asks about the first line that doesn't follow from the one before.
Why it happens
Dividing both sides by the same number is a normal move in solving equations, as long as that number isn't 0. With a number like 5, you know it isn't. With x, you don't know yet, and x = 0 is one of the possibilities you're trying to find. Dividing by x quietly assumes x isn't 0, so that solution disappears without any warning.
Paul Dawkins's list of common algebra errors works through 2x² = x as an example: canceling an x leaves only x = 1/2 and misses x = 0. The same list has a “proof” that 1 = 2 whose hidden mistake is dividing by a − b when a = b, which means dividing by 0.
How to catch it yourself
Before you divide both sides by anything, ask whether it could be 0. If it contains the variable, it could, so collect everything on one side and factor instead. Then point at what you divided by and ask yourself:
Is there a value of x you can't divide by?
It also helps to count solutions. An equation whose highest power is x² can have up to two. If you found only one, look back at the step where something with x in it disappeared.
Try these
Find every solution.
- x² = 7x
- 3x² = 12x
- (x − 2)(x + 1) = 4(x − 2)
Answers
- x = 0 or x = 7
- x = 0 or x = 4, because 3x(x − 4) = 0. Dividing both sides by 3 is fine, but dividing by x would lose x = 0.
- x = 2 or x = 3, because (x − 2)(x + 1 − 4) = 0, which is (x − 2)(x − 3) = 0
How alai helps
In Homework help, when you pause after writing a step, Ally reads it. Ally follows each line and asks about the first one that doesn't follow from the line before, with a hint that stops one step short. Ally also remembers what tripped you up in past sessions, so a later session can start from there.