Algebraic Rules of Thumb

Here are some general algebraic "rules of thumb", with some examples of how they are used:

Important: When working with equations, you should be aware that an = sign can mean several different things.

Do the Same Thing to Both Sides of an Equation

From one perspective, mathematics is all about moving from statements that we believe to be true to other statements that we can know to be true (assuming that our original beliefs are correct).  Most of the time, such statements are expressed in terms of equations.  An equation is true if both sides express the same value, although they are written differently.  As long as we "do the same thing" to both sides, the resulting values will be the same, so we will again obtain a true equation.  

From a functional perspective, we must apply the same function to both sides of an equation.  The challenge of Algebra is to know what function to apply when, and how to simplify the result correctly.  The next four rules discuss the how, while the next two discuss the what and when.  The rule after that discusses how to check your work.

When Substituting, Use Parentheses

Parentheses are important to help us avoid confusion, either for ourselves or others.  In particular, it is important to use them when substituting one expression into another.  See what can happen, if we do not use parentheses:

1.4 + 6-2 - 5-22

While we might know what we mean, if we gave this to someone else to simplify, they would naturally interpret all of the minus signs as subtraction, and (following the correct order of operations) would simplify this as: 

1.4 + 4 - 5 - 4 = 5.4 - 5 - 4 = .4 - 4 = -3.6

It is a good habit to actually say what mean (and mean what we say), which in this case would be:

1.4 + 6(-2) - 5(-2)2 = 1.4 - 12 - 5·4 = -10.6 - 20 = -30.6

z = 3x + 2 - 1 = 3x + 1

instead of:

z = 3(x + 2) - 1 = 3x + 6 - 1 = 3x + 5

In case you are not convinced that the second equation is correct and the second is incorrect, we can verify our equation by plugging-in values.  If x = 1, then y = 1 + 2 = 3, and z = 3·3 - 1 = 8.  The first equation gives z = 3·1 + 1 = 4, but the second gives z = 3·1 + 5 = 8.  This proves that not using parentheses definitely gave an incorrect result, and provides evidence that using parentheses helps to avoid mistakes.

In general, it never hurts to put in too many parentheses.  You can drop them later, if it cannot possibly lead to confusion or ambiguity.  So always surround any expression being substituted into another with parentheses.  

Notice: Since "doing the same thing to both sides" is really functional evaluation, we should remember to use parentheses here, too.  For example, if we divide both sides of:

x + 2 = 2y

by 2, if we forget to use parentheses, we will get:

x + 2/2 = 2y/2 or x + 1 = y 

which is clearly wrong, (by plugging in; x = 2 and y = 2 works in the original equation, but not this one) instead of:

(x + 2)/2 = 2y/2 or x/2 + 1 = y 

Alternatively, if you use fraction notation, since the parentheses are implied, you could write:

Follow the Assumed Order of Operations

Whenever you try to simplify a mathematical expression, you must always follow the correct order of operations.  That is because we are lazy; instead of putting in enough parentheses so that the process of simplification is unambiguous, we establish conventions on where the parentheses are assumed to be, and then do not bother to write them in.  This means that following the correct order of operations is really about knowing where parentheses are assumed to be.

Specifically, we should always perform:

and, in the case of ties (e.g., a sequence of products and quotients):

Note: A minus sign in front of an expression, such as -x or -2 is considered as multiplication by -1, and so occurs before addition, but after powers.

For example, if we put in all the implied parentheses in the expression:

-5x2 + 4xy22 - 12 - z

we would start with powers, working left-to-right:

-5(x2) + 4x((y2)2) - 12 - yz/x/3

then the products:

((-5)(x2)) + ((4x)((y2)2)) - 12 - (((yz)/x)/3)

and finally the sums and differences:

(((((-5)(x2)) + ((4x)((y2)2))) - 12) - (((yz)/x)/3))

You can see why we would rather not always write all these parentheses, but it is important to remember that they are there.  

You should be aware that some operations, like powers, quotients and roots, imply parentheses by the notation itself:

However, in order to maintain the correct order of operations, we must remember to explicitly put in these implied parentheses, when:

For example, if you punch in an expression like without putting in the parentheses yourself (i.e., as Ö1+3+2^1+2), the calculator will probably give you 1 + 3 + 2 + 2 = 8 as a result, instead of .

Likewise, there is a rule of exponents that says "a reciprocal in an exponent is like a root".  This means that we can rewrite as .  However, if we forget to put in the implied parentheses, we would get , which is clearly not the same (which you can see by plugging-in values).

To summarize, you must always remember where the implied parentheses belong (due to order of operations and/or notation) and be able to put them in or take them out, whenever necessary, to correctly simplify an expression.  As with the earlier rule on parentheses, it never hurts to put in too many parentheses.

Combine Like Terms

This widely applicable rule is based on the old adage, "You can't add apples and oranges."  For example, if you add 2 apples and 5 apples you get 7 apples; but if you add 2 apples and 5 oranges you still only have 2 apples plus 5 oranges.  That is, unless you put both in the blender to make juice; then you might get  1 cup of juice from the apples and 3 cups of juice from the oranges to get 4 cups of juice in total.

This principle applies in algebra in a few different ways.  For example:

If you keep your eyes open, you may even discover how even more mathematical techniques can be thought of as applications of this rule-of-thumb.

To Simplify, Work Inside- Out

In mathematics, we use parentheses to help us read complicated expressions properly.  Sometimes the parentheses are not written out explicitly (even though they are implied by the usual order of operations), but we will put them in explicitly to either:

In either case, whenever we want to simplify a mathematical expression, it is a good idea to work "inside-out"; that is:

For example, if we know that f(x) = 3(x - 1) and t = 2, then we would simplify the expression:

(s + 1)f(s + 1) + 4(t - 5) + (s + t - 1)3t - 4

by first putting plugging-in for t:

(s + 1)f(s + 1) + 4(2 - 5) + (s + 2 - 1)3·2 - 4

Since the power includes a set of implied parentheses around the exponent, and around the product, this is the most deeply nested part of the expression, so we simplify it first, to obtain:

(s + 1)f(s + 1) + 4(2 - 5) + (s + 2 - 1)2

Then, the expressions s + 1, 2 - 5, and s + 2 - 1 are all at the same level of parentheses.  The first cannot be simplified (since we can only combine like terms, we cannot add a variable with a constant), the second becomes -3, and the last (by combining like terms) becomes s + 1, so we are left with:

(s + 1)f(s + 1) + 4(-3) + (s + 1)2

Notice that to eliminate the parentheses, we must do something different in each case:

Now we are left with (3s2 + 3s) + (-12) + (s2 + 2s + 1); combining like terms yields 4s2 + 5s - 11).  

To Cancel, Do the Opposite

In Algebra, every operation is paired with its "opposite":

These are "opposites" in that they have the "opposite" effect on "inputs" and consequently "cancel" out each other.  For example, the opposite of "adding 3" is "subtracting 3".  In functional terms, these are inverse functions.

When trying to simplify an equation, we really only have two options:

We usually want to solve an equation for a variable.  Assuming that both sides are simplified as much as possible, and that there is no other equation available to plug-in, we are only left with the second option.  In this case, to solve the equation generally requires that we "cancel" from one side to leave the variable by itself, so we must apply the same operation to both sides which is "opposite" of the one that is in the equation near the variable.  

For example, to solve x + 3 = 7, we should subtract 3 from both sides (since this is the opposite of the operation "add 3" that is applied to x), which gives x + 3 - 3 = 7 - 3.  Once we simplify both sides, we are left with x = 4.  While from a purely mechanical point of view, this can be described as "moving the 3 to the other side", this perspective obscures the basic ideas involved.  

If you have become accustomed to thinking in this mechanical way, you should try to adjust to thinking from this functional perspective.  Instead of being an endless maze of different procedures and arbitrary rules, you will eventually discover that the process of Algebra is largely simply alternating between two basic steps: substituting/simplifying, and canceling.  The main challenge is to know which to do when.  This question is partially answered by the next "rule of thumb".

To Solve, Work Outside-In 

We can solve any equation in which the variable appears only once, by applying the same basic procedure:

For example, starting from the variable x, the operations (i.e., functions) applied to it on the left side of are, in order:

Therefore, to solve for x, we simply apply the opposite (i.e., inverse) operation in the reverse order, that is, working from the "outside-in":

simplifying both sides after each step, to obtain:

When the variable appears more than once, you may need to subtract or multiply by expressions involving the variable to get all the occurrences of the variable on one side, but the principle is the same: apply an "opposite" operation on both sides to cancel out an unwanted expression from one side of an equation.

To Check, Plug-In 

Since an equation usually is supposed to hold for all possible values of its variables, it is impossible to prove that an equation is correct by simply plugging-in (this would take forever).  However, if an equation is incorrect, it is usually incorrect for lots of different values of its variables.  This means that we can provide evidence that an equation is correct or incorrect by plugging-in specific values.  Specifically, if you plug-in values and:

For example, many students often want to use an incorrect rule for adding fractions to obtain an equation like ; however, if we plug-in x = 1, we get (that is, 2 + .5 = 3/3 or 2.5 = 1), so that the original equation is clearly incorrect.  

On the other hand, if  , the quadratic formula says that .  If we take a = 2, b = 5, and c = -3, then we get .  Simplifying, gives x = -3 or x = 1/2.  Plugging-in x = -3 gives , (that is, 18 + (-15) - 3 = 0) which is true.  This gives evidence that the quadratic formula is probably correct.  Note: We can also give a purely algebraic explanation why it always works.

 

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