Sometimes we'll need to solve an equation that has a funky answer, like 10 = 8 or y = y. This doesn't necessarily mean that we did anything wrong; it might very well mean that all or no numbers work. Here are some of these equations.
All Real Numbers Example
Solve 3x + 24 = 3(x + 8) for x.
 | |
 | distribute the 3 |
 | subtract 24 from each side |
 | |
 | divide each side by 3 |
 | well duh! |
| x = all real numbers |
This means that any number we choose for x will make the equation true. We should verify that this is the correct answer by doing just that: picking a few different numbers and seeing if they work. Let's pick easy numbers like 1 and 2.
 |  |
 |  |
 |  |
See, it all works.
BTW, did you notice that when we distributed the 3 in 3(x + 8) at the beginning of the problem, the expressions on each side of the equal sign were exactly the same?
No Solution Example
Solve 5 – 6y = 2(-3y) + 1 for y.
 | |
 | multiply 2(-3y) |
 | add 6y to each side |
 | wait,  |
| no solution |
This equation doesn't work. Since 5 ≠ 1, there is no number we can substitute for y to make this equation true.
Unfortunately this one is harder to verify, since it would be impossible to check that every number in the universe does not work. The best way to make sure the answer is correct is to redo the problem.
Example 1
If we solve an equation and get 6 = 10 in the very last step, is the answer all real numbers or no solution? |
Example 2
Solve for x in the following equation: 4x + 10 = 2(2x + 5) |
Example 3
Solve for x in the following equation: 3x + 2 = 3x – 8 |
