Solving in Several Steps

In this lesson you will: Undo operations in the reverse of the order they were applied; Clear parentheses and fractions before you start isolating the variable; Check a solution by substituting it back into the original equation.

Work backwards out of the expression

An equation like 4x-7=21 was built by taking x, multiplying by 4, then subtracting 7. To get back to x, undo those in reverse: add 7 first, then divide by 4.

That reversal is the whole method. The order of operations tells you how an expression was assembled; solving takes it apart from the outside in.

Clear the clutter first

Two things are worth removing before you isolate anything: parentheses and fractions.

Parentheses go by distributing. Fractions go by multiplying every term by the least common denominator — which usually turns an ugly equation into an easy one in a single line.

Common mistake — Multiplying only some terms by the LCD

To solve (x)/(3)+(1)/(2)=(5)/(6), a student multiplies by 6 and writes 2x+1/2=5.

It is easy to see why. The eye goes to the terms that obviously have denominators, the 1/2 gets skipped in the rush, and the line still looks like progress.

But multiplying one side of an equation means multiplying every term on that side. Missing one destroys the balance, and everything after it is wrong no matter how careful the algebra looks.

How to catch it: Count terms before and after. If the equation had four terms, it still has four terms — each one changed. Here the correct line is 2x+3=5, giving x=1.