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

Mathematics 1010 online

Linear Equations

An equation is linear if it can be written in the form

$\displaystyle ax=b $

where $ x $ is the variable and $ a $ and $ b $ are constants. The crucial part of this definition is the phrase can be written, because most linear equations do not occur in as simple a form as above.

Solutions of a Linear Equation

Let's first ask what kind of solutions are possible for a linear equation. There are three cases:

  1. A Unique Solution

    Suppose $ a\neq 0 $. In that case there is one and only one solution, namely

    $\displaystyle x = \frac{b}{a}. $

    For example, the equation $ 3x=6 $ has the solution $ x=6/3=2 $, and there is no other solution. We say that the equation has a unique solution. This is by far the most frequent and the most important case.

  3. No Solutions

    Suppose $ a = 0 $ and $ b\neq 0 $. In this situation we have an equation like $ 0x = 1 $ and clearly there is no solution.  

  4. Infinitely Many Solutions

    Suppose $ a = 0 $ and $ b = 0 $. In this situation we have the equation $ 0x=0 $ and this is clearly true for all values of $ x $. There are infinitely many solutions.

Clearly there are no other possibilities, and we note the important fact that a linear equation may have none, one, or infinitely many solutions. It is not possible, for example, that a linear equation has two solutions.

The crux of solving a linear equation is to recognize that the equation is linear, and to convert it to the above simple form. In the remainder of this page this process will be illustrated for increasingly complex equations.

The material on this page illustrates three of our principles:

Below you will see several examples of how an equation whose linearity is not obvious can be converted to a clearly linear equation. The examples cover most or all of the techniques required for this class, but of course the list is nowhere near being exhaustive.

Those are the principles, in a nutshell. There are of course a great many more examples and exercises in your textbook.

Nonlinear Equations

It is important to realize that not all equations are linear. For example, the equation

$\displaystyle (x-2)(x-3) = 0 $

has the solutions $ x=2 $ and $ x=3 $, and it is not linear. Since it has two solutions, which is not possible for a linear equation, we can tell without trying that it cannot be converted into $ ax=b $.