# Equations

An equation is a statement claiming that two algebraic expressions are equal. This is indicated simply by placing an equals sign ‘=’ between the expressions.

## Checking equations

Recall that you evaluate an expression by substituting specific numbers for each of the variables; when you work it out, the result is another specific number (assuming that it's defined). You evaluate an equation in much the same way, by substituting specific numbers for each of the variables; however now, when you work it out, you will get a numerical statement that is either true or false (assuming again that it's defined). This is called checking the equation for those values.

For example, consider the equation 2x − 4 = 3x − 9. If you substitute x = 4 (say), then you get this result:

• 2x − 4 = 3x − 9 — original equation;
• 2(4) − 4 = 3(4) − 9 — substitute x = 4;
• 8 − 4 = 12 − 9 — work out 2 · 4 = 8 and 3 · 4 = 12;
• 4 = 3 — work out 8 − 4 = 4 and 12 − 9 = 3;
• False — since in fact 4 > 3.
On the other hand, if you substitute x = 5, then you get this result:
• 2x − 4 = 3x − 9 — original equation;
• 2(5) − 4 = 3(5) − 9 — substitute x = 5;
• 10 − 4 = 15 − 9 — work out 2 · 5 = 10 and 3 · 5 = 15;
• 6 = 6 — work out 10 − 4 = 6 and 15 − 9 = 6;
• True — since in fact 6 = 6.
So the equation 2x − 4 = 3x − 9 is false when x = 4 but true when x = 5.

For another example, consider the equation t = 1/t. Let me try t = 2, t = 1, and t = 0.

• t = 1/t — original equation;
• 2 = 1/2 — substitute t = 2;
• False — since in fact 2 > 1/2.
So the equation is false when t = 2.
• t = 1/t — original equation;
• 1 = 1/1 — substitute t = 1;
• 1 = 1 — work out 1/1 = 1;
• True — since in fact 1 = 1.
So the equation is true when t = 1.
• t = 1/t — original equation;
• 0 = 1/0 — substitute t = 0;
• Undefined — since 1/0 is undefined.
So the equation is meaningless (its truth value is undefined) when t = 0.

For the most part, a meaningless equation is just as good or bad as a false one, and in fact some mathematicians would count this equation as false when t = 0.

## Solutions

A solution to an equation is an assignment of values for the variables in the equation such that the equation becomes true when evaluated at those values. For example, we've seen above that x = 5 is a solution to 2x − 4 = 3x − 9, while x = 4 is not a solution of that equation. Also, we've seen that t = 1 is a solution to t = 1/t, while t = 2 and t = 0 are not solutions (albeit for different reasons).

An equation may have just one solution, many solutions, or none at all. As it turns out, x = 5 is the only solution to 2x − 4 = 3x − 9. However, t = 1 is not the only solution to t = 1/t. This is because t = −1 is also a solution:

• t = 1/t — original equation;
• −1 = 1/(−1) — substitute t = −1;
• −1 = −1 — work out 1/(−1) = −1;
• True — since in fact −1 = −1.

For an example of an equation with no solutions, consider x = x + 1; no matter what real number x is, x will in fact always be less than x + 1, so nothing could possibly be a solution to this equation. For an example of an equation with many solutions, consider the equation x = |x|; one solution is x = 0, but in fact any positive number could be used instead. (On the other hand, no negative number gives a solution.) Or consider the equation x = x. No matter what real number you substitute for x, this statement will obviously be true! Finally, consider the equation 1/x = 1/x. Obviously this statement is true whenever its defined, but x = 0 is not a solution, because the result is undefined in that case.

So in summary, an equation might have no solutions, one real number as a solution, a few solutions, a whole range of solutions, every real number as a solution, or every real number with one or a few exceptions. Pretty much anything is possible if you pick the right equation!

## Equivalent equations

Two equations (or inequalities, or compound statements) are equivalent if they have the same solutions. That is, any assignment of values to variables that makes either statement true will also make the other statement true.

For example, since x = 5 is the only assignment that will make

2x − 4 = 3x − 9
true, and it's also (obviously) the only assignment that will make
x = 5
true, these two equations are equivalent. Similarly, the equation
t = 1/t
is equivalent to the compound statement
t = 1 or t = −1,
because in each case the only solutions are t = 1 and t = −1. Also, the inequality
2x − 4 > 3x − 9
is equivalent to the inequality
x < 5,
because in each case, the solutions are given by assigning x to any value less than 5.

The book doesn't introduce a symbol to describe equivalence of statements, but you can use ‘⇔’ if you wish. (This symbol is usually read aloud as ‘if and only if’.) For example, the equivalences above can be symbolised thus:

• 2x − 4 = 3x − 9  ⇔  x = 5;
• t = 1/t  ⇔  t = 1 or t = −1;
• 2x − 4 > 3x − 9  ⇔  x < 5.
Now, you don't really need this symbol; as you solve an equation, you usually just list equivalent equations in a column, from your original equation to the final answer. So as far as I'm concerned, you don't have to learn this symbol. However, I do want to make the point that you should not use an equals sign! For example, this would be quite wrong:
2x − 4 = 3x − 9  =  x = 5.  (WRONG!)
In this problem, you do not want to say that 3x − 9 is equal to 5.

## Solving equations

For purposes of this course, we can consider an equation with one variable solved if it is one of the following:
• An equation with the variable alone on the left and a (defined, simplified) constant on the right;
• Any number of such equations joined by ‘or’, with no repetitions, listed in increasing order;
• The statement ‘True’ (which is always true); or
• The statement ‘False’ (which is always false).
Not every Algebra problem can be put in one of these forms; aside from the possibility of an inequality or an equation with two or more variables (both of which I'll discuss later), if you take Intermediate Algebra or (worse) Trigonometry, you'll run across equations in only one variable whose solution sets are still more complicated. However, all the equations in one variable that we'll study in this course (and many that you'll come across later too) will have solutions in the above forms.

I have put more detail in the specifications here than you really need. For example, it doesn't really matter if the variable is on the left or the right, and you don't always have to give an ‘or’ statement in increasing order. But the forms as I've described them are the most common.

## Finding equivalent equations

The easiest way to turn an equation into an equivalent equation is to replace one side (or both) with an equivalent expression. For example, the expression 2(x + y) is equivalent to the expression 2x + 2y, so the equation
2(x + y) = 7x
is equivalent to the equation
2x + 2y = 7x
(and of course there's nothing special about the 7x here).

This trick has nothing to do with equations as such, and it works just as well for inequalities (or any other statement one might make about real numbers). The big idea for solving equations specifically is this: Do the same invertible operation to both sides. An invertible operation is any operation on a real number that is always defined and that has another (inverse) operation that will always turn the result back into the number you started with.

For example, you can add 5 to any number, and you get back where you started if you then subtract 5, so adding 5 is an invertible operation. Or you can add x to any number, and you get back where you started if you then subtract x, so adding x is an invertible operation. In fact, you can add any defined expression you like, and you get back where you started if you then subtract that same expression, so adding any defined expression is always an invertible operation.

Other invertible operations include subtracting any expression, multiplying by any expression known to be nonzero, and dividing by any expression known to be nonzero. For multiplication and division, it's important that you know that the expression is nonzero; usually, this means that you can only multiply or divide by constants. In general, you can't divide by x, because if x is 0, then this operation is not defined. And you can't multiply by x, because if x is 0, then this operation is not invertible. (However, sometimes you're in a situation where you know that x can't be 0; then it's OK to multiply or divide by x.)

Finally, you can always turn an equation into an equivalent equation by swapping the two sides. This may seem silly, but it's sometimes nice to do.

In summary, here's a list of techniques for solving equations:

• Simplify either side (or both);
• Add the same expression to both sides (and then simplify them);
• Subtract the same expression from both sides (and simplify);
• Multiply both sides by the same nonzero constant (and simplify);
• Divide both sides by the same nonzero constant (and simplify);
• Swap the sides.
(In this list, I've included how you almost always want to replace each side with an equivalent expression by simplifying it.)

I'll discuss later another technique useful when you have an absolute value in an expression. You'll learn some others in Intermediate Algebra. There are always more techniques, some of which are still being discovered.

## Solution sets

The solution set of a statement about real numbers is that part of the real line where the statement's solutions are. The graphs above are pictures of such solution sets. But it's also helpful to have an algebraic notation for the solution set. Here are the notations for the solution sets of the equations that appear in this course:
• If there is just one solution, then put that solution in curly braces. For example, if x = 4 is the only solution, then the solution set is {4}.
• If there are a few solutions (but only a finite list of them), then put that list in curly braces, separated by commas. (You can list them in any order, but increasing order is usually easiest to understand.) For example, if y = −2 or y = 4 or y = 6, then the solution set is {−2, 4, 6}.
• If every real number is a solution, then the solution set is the set of all real numbers, written R.
• If no real number is a solution, then the solution set is {}, called the empty set.
Strictly speaking, you should say that {4} is the solution set for x of the equation x = 4. Similarly, {−2, 4, 6} is the solution set for y of y = −2 or y = 4 or y = 6.

Usually, it's best to write an answer as a simple statement; this is more concrete, and it makes clear what variable you're discussing. You usually want sets only in more abstract settings.

## Checking solutions

You already know how to check whether a certain assignment of variables is a solution to an equation (or inequality). But you've also been learning techniques for solving equations (and you'll learn more, and in more depth, over the next couple of weeks), so you might wonder why you ever need to check. Just solve the equation; then you'll know what the solutions are!

Actually, there are several good reasons to check solutions:

• Some equation-solving techniques do not yield equivalent equations! These techniques sometimes yield extraneous solutions, which appear to be solutions but are not really, so you have to check all of your solutions when you use these techniques. You won't have to deal with these until next term, but it's worth keeping in mind.
• Sometimes the equation that you set up for a word problem doesn't take account of all of the relevant information in a problem. (For example, a length should usually be positive, but you might not notice this when you decide to call it l, which in principle might stand for a negative number.) You should always check your answers to word problems to ensure that they make sense in the original context.
• If you're only interested in one or a few possible solutions, then probably it's easier simply to check them than to solve the equation from scratch.
• If you have a multiple-choice exam (not in this course, but perhaps elsewhere), then it might be a lot easier than you think— if you only have to check the offered solutions. (A ‘none of the above’ option can sometimes cause problems, however.)
• If you're not sure whether you solved an equation correctly (do you ever make mistakes?), then checking your solutions is a quick way to verify your answer.
The best advice that I can give for checking solutions is this: Always check your solutions in the original problem! If you only check in a step halfway through, then you'll never catch any mistakes in the first half of your solutions. On the other hand, if you do know that you made a mistake but you're not sure where, then it can help to check your solution in various intermediate steps until you find the first place that it seems to work; the mistake must have come just before that step.
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