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Solving Linear Equations

One type of equation that often arises in mathematics is the linear equation.  These equations are called linear because in the two dimensional case, the graph of a linear equation is a line (see the tutorial entitled Equations and Graphs of Lines).  A linear equation is any equation that is made up of numbers and variables in which no variable is raised to a power other than one.  Here are some examples:



Linear equations can include any number of variables.  In this tutorial, we will focus only on the method for solving linear equations which have one variable.

The idea behind solving a linear equation for one variable is simply to apply rules of algebra until the variable is isolated on one side of the equation.  This can be done by following these steps:

            1) Simplify both sides of the equation, removing any parentheses by distributing
                 and combining like terms whenever possible.
           
            2) Next, collect all the constant terms to one side, collect all the terms containing
                 the variable on the other and then combine the like terms.
           
            3) Finally, divide both sides by the coefficient of the variable (or more simply in
                 some cases, multiply by the reciprocal of the coefficient of the variable).

If you follow these steps, you obtain the value the variable must have in order for your original equation to be true.  The most important thing to keep in mind when solving an equation is this:

 

In order to preserve equality, what you do to one side of an equation must be done to the other side as well.

This is the cardinal rule of algebra. 

Let’s apply the cardinal rule to the third example from above and solve for x in that equation.

Example

Solve for x. (In other words, find the value of x that would make the right and left hand sides of the equation equal).


Solution

First we simplify both sides.  Notice that in order to remove the parentheses on the left hand side, we must distribute the coefficient of –1 and on the right hand side we must distribute the coefficient of 2.  What we get is:

Now combining like terms gives us:

Next we add 3 to both sides so that constant terms no longer appear on the left hand side and we subtract 2x from both sides so that terms with the variable x no longer appear on the right hand side.  What we are left with is this equation:

It is important to note that while this equation looks different from the one we started with, it is equivalent in that a solution to this equation is a solution to the original equation.

Now let’s combine like terms to get:

Dividing both sides by 3 which is the coefficient of x, gives us the equation:

Or more simply:

Let’s confirm that x = -4 is the correct solution to the equation.

Our original equation was:

If we substitute the value of -4 for x, we have this:

                                                -4 – (3 – 4(-4))           =  2(-4 – 3) – 9
                                                -4 – (3 + 16)               =  -8 – 6 – 9
                                                -4 – 19                        =   -14 – 9
                                                -23                              =  -23

In other words, when x = -4, our equation is a true statement.
Thus, x = -4 is the solution of the equation.