# On the origin of algebra problems

As you are solving a math problem, have you ever wondered how textbook authors come up with algebra problems that have simple answers?

Just as you use inverse functions to solve an algebra problem, problem authors reverse the solution process to create a problem: they start with the answer.  From there they use the same principles that are used to solve an algebra problem, except instead of simplifying the equation at each step, they seek to make it more complicated by substituting a series of equivalent expressions and/or performing a series of operations to both sides of the equation.

Suppose you want the answer to be

$x=3$

Pick something to do to both sides, say multiply by 2:

$2x=6$

Do something else to both sides, say add 4:

$2x+4=10$

I could add 4x to both sides… why 4x?  Because I felt like it:

$4x+2x+4=4x+10$

I could rewrite the 4 on the left side as 2+2:

$4x+2x+2+2=4x+10$

Now to add another step to the solution process, I’ll factor part of the left side:

$4x+2(x+1)+2=4x+10$

And factor part of the right side too:

$4x+2(x+1)+2=2(2x+5)$

And voila!  An algebra problem which is guaranteed to have x = 3 as a solution.  Try creating several problems this way yourself.  Now that you know how to create your own algebra problems, and once you get some practice creating them, you may find it easier to solve algebra problems.