Using Balance

The primary rule when solving an algebra equation is: you can do whatever you want to one side of an equation as long as you do the same thing to the other side.  So, you can add something to both sides of an equation, multiply or divide both sides of an equation by the same number, square or square root both sides of an equation.  Every operation is fair game as long as you perform the same operation on the other side of the equation to keep the equation in balance.

Typically, we manipulate one side of an equation in order to isolate a variable (or reorganize an equation to graph it).

But, tests sometimes use the "balance" rule to make an algebra problem look harder than it is.

Some algebra problems look complicated because there are a lot of terms or polynomials that are divided or squared.  However, many tests like to use questions where they “do something” to a polynomial and all you have to do is keep the equation in balance and “do the same thing” to the other side of the equation.  These problems are easier than they look.

So, often a problem will give you an equation and ask you to solve another equation, but if you look hard you realize that the second equation is identical to the first equation with one additional operation.

For instance, a problem might ask: 

If $x + 7 = 30$, what is $(x + 7)^2$?

In this case, they took $x+7$ and just squared it.  So, to find the answer, you do the same thing to the other side of the equation.

$$\eqalign{\text{If } x+7 &=30 \text{ , then }\\(x+7)^2&=30^2\text{ , so }\\ (x+7)^27&=900}$$

In these cases you don't even have to solve for x, you solve the problem by keeping the equation in balance!  This process works no matter what operation is applied to the equation.

Example: If $y-40=15$, what is $3(y-40)$?

If you look closely at the problem, you can see that they just took the original expression and multiplied it by 3.

$$\eqalign{\text{If } y+40 &=15 \text{ , then }\\3(y+40)&=3(15)\text{ , so }\\3(y+40)&=45}$$

Obviously, this trick doesn't work all the time.  But when you're working through a test, it's always helpful to keep your eyes out for little tricks like this.  They can save you a lot of time and effort.  

Bottom line: never solve for a variable if you don't have to!