Whatever you do to one side you must do to the other. That single rule is the whole method.
An expression and an equation are different
3x + 5 is an expression with a value for every x. 3x + 5 = 20 is a claim that can be true or false.
Take away the 5 first, then divide by the 3. The last thing done is the first thing undone.
| Step | Equation | What was done to both sides |
|---|---|---|
| 1 | 3x + 5 = 20 | The equation as given |
| 2 | 3x = 15 | Subtracted 5 from both sides |
| 3 | x = 5 | Divided both sides by 3 |
| 4 | Check: 3(5) + 5 = 20 | Substitute back into the original |
If 3x + 6 = 15 and you divide by 3, you get x + 2 = 5, not x + 6 = 5.
Letters on both sides
Collect all the x terms on one side and all the numbers on the other, using the same balance rule.
For x = 4 in 5x − 3 = 2x + 9, the left is 17 and the right is 17. This check catches every arithmetic slip.
Turning a problem into an equation
Name the unknown first and say what it stands for, then write the sentence in symbols.
2x = 8 is not an answer, because x has not been found yet. Read your last line and ask whether it gives the value of x.
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