Study Guide

Basic Algebra - Solving Funky Equations

Solving Funky Equations

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.

3x + 24 = 3(x + 8)
3x + 24 = 3x + 24distribute the 3
3x + 24 - 24 = 3x + 24 - 24subtract 24 from each side
3x = 3x
3x/3 = 3x/3divide each side by 3
x = xwell 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.

3(1) + 24 = 3(1 + 8)3(2) + 24 = 3(2 + 8)
3 + 24 = 3(9)6 + 24 = 3(10)
27 = 2730 = 30

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.

5 - 6y = 2(-3y) + 1
5 - 6y = 6y + 1multiply 2(-3y)
5 - 6y + 6y = -6y + 1 + 6yadd 6y to each side
5 = 1wait, 5 (not equal to) 1
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.

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