Department of Mathematics --- College of Science --- University of Utah

Mathematics 1010 online

Equations and Identities

An equation consists of two algebraic expressions and the symbol $ = $ between them. The algebraic expressions contain variables and constants. If the equation is true for all values of the variables it is called an identity. An example of an identity is

$\displaystyle a+b = b+a. $

This is true for all real numbers $ a $ and $ b $, and it is called the commutative law of addition. A more subtle identity is

$\displaystyle (a+b)^2 = a^2 + 2ab + b^2. $

This is called the first binomial formula. The second and third binomial formulas, respectively, are

$\displaystyle (a-b)^2 = a^2 - 2ab + b^2\qquad\hbox{and}\quad (a+b)(a-b) = a^2-b^2. $

Solving Equations

The focus on this page, however, is on equations that are true only for some values of the variables. There may be several such variables, but to begin with we assume there is only one, and we usually call it $ x $. Figuring out for which values of $ x $ an equation is true is called solving the equation. A value of $ x $ that makes the equation true is called a solution of the equation. For example, the equation

$\displaystyle 3x+4 = 7 $

is clearly true if $ x=1 $ and so $ x=1 $ is a solution of the equation. It is also the only solution. A more subtle example is provided by the equation

$\displaystyle x^2-5x+6 = 0. $

It's easy to check that $ x=2 $ and $ x=3 $ are both solutions of the equation. It turns out that these are all the solutions.

Principles

The fundamental principle of equation solving is based on the fact that after applying the same operation on both sides of the equation the solutions of the original equation are also solutions of the new equation. Think of the equation as one of those old fashioned scales that balance an initially unknown weight with a combination of known weights. If everything is in balance and you do whatever you do on both sides everything will still be in balance.

Simply put, the fundamental principle of equations solving is

To solve an equation figure out what bothers you the most at the moment and get rid of it by applying a suitable operation on both sides of the equation.

For example, consider the above mentioned equation $ 3x+4 = 7. $ We want to solve it. This means we want to obtain another equation of the form $ x=\ldots $ where the right hand side of the equation does not involve $ x $. Well, $ x $ is not by itself in the original equation. It is multiplied with 3 and 4 has been added. Both the 3 and the 4 bother us. We could get rid of them in either sequence, but it's simpler to get rid of 4 first, by subtracting 4 on both sides of the equation. Since $ 3x+4-4=3x $ and $ 7-4=3 $ this gives the new equation.

$\displaystyle 3x=3. $

We are still bothered by the factor 3. So we divide by 3 on both sides. Since $ 3x\div 3 = x $ and $ 3\div 3 = 1 $ we obtain

$\displaystyle x=1. $

Of course we knew that all along, but this simple example illustrates how all equations (at least in this class) are solved.

In the literature, like in your texbook, the simplicity and power of the fundamental principle is obscured by the fact that there is a long list of special cases. For example, just for linear equations our textbook lists: solving linear equations in standard form, linear equations in non-standard from, linear equations involving fractions, linear equations involving decimals, and linear equations--special cases. The issue is further confused by giving various names to "applying the same operation on both sides". For example, in our textbook there is the "addition property of equality" (meaning you can add the same term on both sides), the "multiplication property of equality" (meaning you can multiply with the same non-zero factor on both sides), etc.

This is like having a city guide that contains sections on how to walk along 13th East, how to walk along 7th East, how to walk across First South, and so on. Actually, all you need to know is how to walk. Similarly, to solve equations, all you need to understand and appreciate is the above mentioned principle. Once you do that solving equations is just a matter of practicing and gaining experience.

There is a subtlety that is hard to appreciate at first. "Doing the same thing on both sides" may introduce additional solutions (called spurious in this context). For example, consider the equation

$\displaystyle x=1. $

Squaring on both sides gives the new equation

$\displaystyle x^2=1. $

since $ 1^2=1 $. The value $ x=1 $ is still a solution of the new equation, but since $ (-1)^2=1 $, the value $ x=-1 $ is also a solution of the new equation. However, it clearly contradicts (is not a solution of) the original equation.

This gives rise to an auxiliary principle of equation solving:

After solving an equation check your solutions by substituting them in the original equation.

Checking your answers carefully not only eliminates spurious solutions but it also helps guard against errors.

Two equations that have the same solutions are called equivalent. Ideally one would like to obtain in all cases a string of equivalent equations that end with the final equation $ x=\ldots $, but this is not always practical. It is more effective not to worry about equivalence, accept the possibility of spurious solutions, press on determinedly, and sort things out by checking at the end of the process.

Your textbook contains many examples of solving equations. Look at them, but don't loose sight of the overall picture and the above principles. We will see them in action as we work through the course.