Solutions of Algebraic Equations
Up until now, we have just been talking about manipulating algebraic expressions. Now it is time to talk about equations. An expression is just a statement like
2x + 3
This expression might be equal to any number, depending on the choice of x. For example, if x = 3 then the value of this expression is 9. But if we are writing an equation, then we are making a statement about its value. We might say
2x + 3 = 7
A mathematical equation is either true or false. This equation, 2x + 3 = 7, might be true or it might be false; it depends on the value chosen for x. We call such equations conditional, because their truth depends on choosing the correct value for x. If I choose x = 3, then the equation is clearly false because 2(3) + 3 = 9, not 7. In fact, it is only true if I choose x = 2. Any other value for x produces a false equation. We say that x = 2 is the solution of this equation.
- The solution of an equation is the value(s) of the variable(s) that make the equation a true statement.
An equation like 2x + 3 = 7 is a simple type called a linear equation in one variable. These will always have one solution, no solutions, or an infinite number of solutions. There are other types of equations, however, that can have several solutions. For example, the equation
x2 = 9
is satisfied by both 3 and –3, and so it has two solutions.
This is the normal case, as in our example where the equation 2x + 3 = 7 had exactly one solution, namely x = 2. The other two cases, no solution and an infinite number of solutions, are the oddball cases that you don’t expect to run into very often. Nevertheless, it is important to know that they can happen in case you do encounter one of these situations.
Infinite Number of Solutions
Consider the equation
x = x
This equation is obviously true for every possible value of x. This is, of course, a ridiculously simple example, but it makes the point. Equations that have this property are called identities. Some examples of identities would be
2x = x + x
3 = 3
(x – 2)(x + 2) = x2 – 4
All of these equations are true for any value of x. The second example, 3 = 3, is interesting because it does not even contain an x, so obviously its truthfulness cannot depend on the value of x! When you are attempting to solve an equation algebraically and you end up with an obvious identity (like 3 = 3), then you know that the original equation must also be an identity, and therefore it has an infinite number of solutions.
Now consider the equation
x + 4 = x + 3
There is no possible value for x that could make this true. If you take a number and add 4 to it, it will never be the same as if you take the same number and add 3 to it. Such an equation is called a contradiction, because it cannot ever be true.
If you are attempting to solve such an equation, you will end up with an extremely obvious contradiction such as 1 = 2. This indicates that the original equation is a contradiction, and has no solution.
o An identity is always true, no matter what x is
o A contradiction is never true for any value of x
o A conditional equation is true for some values of x