Solving Algebraic Equations
When an equation has a single variable, the solution to the equation is the number that satisfies the equation (i.e. that makes the equation true) when we substitute that number for the variable name wherever it appears in the equation. Consider the following equation:
6x + 13 = 43
This is a relatively easy equation to solve, since we can simply re-arrange the equation to show that six multiplied by x equals thirty, so x must be five. Substituting five for x, we can re-write the equation using simple arithmetic terms as follows:
6 x 5 + 13 = 30 + 13 = 43
An equation like "6x + 13 = 43" is called a linear equation in one variable. Such an equation will usually have just one solution. This suggests one approach that can be used for solving an equation, which is to substitute values into the equation until it becomes true. This approach, which may be referred to as a brute force or trial and error approach, is only practical if the values that might satisfy the equation are members of a finite set of numbers. Even then, the approach is only practical if the number of possible answers is relatively small. If the answer is known to be an integer, such an approach may well work, given enough time. If the answer could be a rational number, or a number with a rational component, the potentially huge number of possible solutions is likely to make the trial and error approach impractical.
Assuming that a trial and error approach is feasible for solving a particular problem, even an incorrect guess can sometimes be useful. It may become clear, for example, that the number we are seeking is either larger or smaller than the unsuccessful candidate value. This being the case, we can reduce the number of possible answers to a much smaller range of values. While a trial and error approach to solving equations may not sound particularly scientific, it may be the only approach possible when there is no established method of solving a particular type of problem. One example of its use is in attempts to break encryption codes using computers.
To encode a message, a standard encryption algorithm is used to scramble the message using a key. An algorithm is essentially a series of mathematical operations that are carried out on the unencrypted message. The process transforms the message in such a way as to render it unintelligible. The key provides a random variable used by the algorithm. The message can be unscrambled by the recipient using the same algorithm, but only if they have a copy of the key. The encryption algorithm itself is often in the public domain, but the key is a very large, randomly generated number known only to the sender and receiver of the message. The only way of reading the message without a key is to try every possible permutation of digits with the algorithm to see which one produces a coherent message. This can keep a number of supercomputers busy for a very long time!
Fortunately for us, there are usually more reliable methods available for solving the kind of algebraic problems we are likely to encounter. Unfortunately, there is no one-size-fits-all solution, so it is a case of determining what kind of problem we are dealing with, and then applying the appropriate procedure in order to solve it. The approach used for solving a particular equation will depend on the type of equation, the kind of expressions involved in the equation, and the nature of the possible values that could be taken by the variables. For certain classes of equation, there are well-established methods for solving them that can be implemented as computational algorithms. In many cases, these methods work equally well using pencil and paper.
Simplifying equations, by definition, makes them easier to solve. The ultimate aim of simplifying the equation in most cases is to get the unknown value on its own on one side of the equation. The other side of the equation should at that point consist either of a single known value, in which case we have already solved the equation, or of a number of known values related by an expression that can be evaluated to give us a result. There are several ways in which we can simplify an equation:
- Evaluate expressions within parentheses (brackets).
- Combine like terms.
- Add some value to both sides of the equation.
- Subtract some value from both sides of the equation.
- Multiply both sides of the equation (by any number except zero).
- Divide both sides of the equation (by any number except zero).
Evaluate expressions within parentheses
If an equation includes an expression that contains bracketed terms, we should attempt to eliminate the brackets. Consider the following example:
3(x + 4) = 24 ⇒ 3x + 3 x 4 = 24 ⇒ 3x + 12 = 24
Having multiplied out the expression 3(x + 4), we can now subtract twelve from both sides of the equation:
3x + 12 - 12 = 24 - 12 ⇒ 3x = 12
Now divide both sides of the equation by three to get:
|3x||=||12||⇒ x = 4|
We have actually used three of the techniques mentioned above to solve this equation, which serves to illustrate that most algebraic problems (even the relatively simple ones) are solved in a number of stages, each of which reduces the problem to simpler terms. Note that in this particular example we could also have eliminated the brackets by dividing each side of the equation by three, and then subtracting four from both sides:
|3 (x + 4)||=||24||⇒ x + 4 = 8|
x + 4 - 4 = 8 - 4 ⇒ x = 4
Combining like terms
One fairly obvious thing to do in order to simplify an equation that has multiple terms is to group any like terms together. Purely numeric terms can be combined, as can any terms that involve the same variable type. Consider the following expression:
3 + 9x + 15 - 2x - 7
The numeric terms here are the numbers three, fifteen and seven (3, 15 and 7). The other terms are both integer multiples of the variable x. We can combine the numeric terms as follows:
3 + 15 - 7 = 18 - 7 = 11
We can also combine the terms involving x:
9x - 2x = 7x
We can therefore rewrite the expression as:
11 + 7x
The addition principle
The addition principle can actually involve either addition or subtraction, since what we are talking about here is either adding some quantity to both sides of the equation, or subtracting some quantity from both sides of the equation, in order to simplify it. By so doing, we will be changing the equation from its original form to some equivalent form that is easier to evaluate. The values of the unknown quantities (i.e. the variables) that satisfy an equation (i.e. that make it true) are known as the solution set. There may be only one such value for a given variable, or there may be many. Whatever we do to the equation in order to simplify it, we must make sure that the solution set remains the same for the new (equivalent) version of the equation as for the original version. By definition, both sides of an equation are equal. In order to preserve that equality, the guiding principle we must adopt is that whatever we do to one side of the equation, we must also do to the other side. Consider the following (very trivial) example:
4 = 4
If we now add the same value to both sides of the equation, it should still be true. Taking the number five as an example, we get the following:
4 + 5 = 4 + 5 ⇒ 9 = 9
We can do the same kind of thing to an equation involving variables. Supposing we have:
y - 3 = 7
If we now add threeto both sides we get:
y - 3 + 3 = 7 + 3 ⇒ y = 10
The idea in these relatively simple equations is to isolate the variable whose value we seek on one side of the equation. Sometimes, rather than adding a quantity to both sides, we might need to subtract it. Suppose we have:
x + 9 = 21
If we want to get x on its own, we need to subtract nine from the left-hand side of the equation, which means that in order to preserve equality between the two sides of the equation we need to also subtract nine from the right-hand side, as follows:
x + 9 - 9 = 21 - 9 ⇒ x = 12
If you find the concept of subtraction being associated with something called the addition principle a little strange, you can think of this instead as adding minus nine to both sides:
x + 9 + (-9) = 21 + (-9) ⇒ x = 12
In both of the examples we have used here, the idea is to isolate the variable on the left-hand side of the equation. We do this by getting rid of the numeric part of the expression on that side, while at the same time maintaining the equivalence of both sides. What we have done is functionally equivalent to moving the unwanted part of the expression from the left-hand side of the equation to the right-hand side, and then changing its sign. Although the idea of moving terms from one side of an equation to the other is perhaps a convenient way of visualising what we have done, we should not lose sight of the fact that what we have actually done is to apply an identical mathematical operation to both sides.
The multiplication principle
The multiplication principle can actually involve either multiplication or division, since we are talking about either multiplying both sides of the equation by some quantity, or dividing both sides of the equation by some quantity, in order to simplify it. As with the addition principle, the idea is to change the equation from its original form to some equivalent form that is easier to evaluate. Consider the following (very trivial) example:
6 = 6
If we now multiply both sides of the equation by the same value, it should still be true. Taking the number seven as an example, we get the following:
6 × 7 = 6 × 7 ⇒ 42 = 42
As with the addition principle, we can do the same kind of thing to an equation involving variables. Supposing we have:
If we now multiply both sides by three we get:
|3 ×||y||= 3 × 14 ⇒ y = 42|
Once again, the idea is to isolate the variable whose value we seek on one side of the equation. Sometimes, rather than multiplying both sides by some value, we might need to divide them. Suppose we have:
3x = 63
If we want to get x on its own, we need to divide the left-hand side of the equation by three, which means that (in order to preserve equality between the two sides of the equation) we must also divide the right-hand side by three, as follows:
|3x||=||63||⇒ x = 21|
Again, if you find the concept of division being associated with something called the multiplication principle a little strange, you can think of this instead as multiplying both sides by one third (i.e. by the reciprocal of three):
1/3 (3x) = 1/3 (63) ⇒ x = 21
Putting it all together
We will come across many problems in algebra for which we will need to apply some or all of the principles outlined above in order to solve an equation. They will work equally well for situations in which there are unknown values on both sides of the equation. Consider the following equation:
6 (x + 4) = 2 (x + 15) + 12 + x
We can start by getting rid of the brackets:
6x + 24 = 2x + 30 + 12 + x
Now combine like terms on the right hand side of the equation:
6x + 24 = 3x + 42
Subtract 3x from both sides:
6x - 3x + 24 = 3x - 3x + 42 ⇒ 3x + 24 = 42
Subtract twenty-four from both sides:
3x + 24 - 24 = 42 - 24 ⇒ 3x = 18
Finally, divide both sides by three:
|3x||=||18||⇒ x = 6|
The methods described above for solving equations can be applied to a broad range of common algebraic problems, including linear equations with a single variable type. There are however more complex types of equation that cannot usually be solved using these methods alone. These include linear equations with multiple variable types, quadratic equations, and various classes of polynomial equation. We have not attempted to cover methods of solving such equations here, as they will be dealt with on a case-by-case basis. A discussion of quadratic equations and their solution, for example, will be dealt with on the page entitled "Quadratic equations".