Solving Systems of Equations

(1) Solving Systems of Equations by Graphing

(2) Solving Systems of Equations by Substitution

(3) Solving Systems of Equations by Elimination

(4) Special Cases

(5) Systems of Equations WORKSHEETS

The solution to a system of equations is the ordered pair (x, y) that makes both equations true. For example (2, 3) is the solution of the system:

2x + y = 7
4x - 2y = 2

Because:

2(2) + 3 = 7

and

4(2) - 2(3) = 2

It is also important to remember that the graph of a linear equation consists of all the points that make that equation true.

Thus to solve a system of equations by graphing you find the point where the two lines intersect. Because this point is on both lines it thus makes both equations true. So the intersection point represpents the solution to the system of equations.

Special Cases

1) The lines are parallel to each other - By definition two lines that are parallel never intersect each other. Thus they have no points in common and as a result we say the answer is "no solution". The best way to confirm this result is to put both equation in slope intercept form (y = mx + b). If the slopes are the same and the y - intercepts are different then the lines are parallel.

2) The lines are the same - Occasionally the two equations will look different but actually be the exact same line. When you graph them you will notice that they are the exact same line. As with before the simplest way to determine this is to put both equations in slope intercept form.

Some Other Things to Consider

1) Graphing by hand is not very precise. Thus occasionally it is difficult to determine the exact solution. If an exact solution is necessary then it would be best to use one of the other methods described below.

2) If you are having trouble graphing then check out some of the links below. I will be designing a lens on graphing, but it will be a few months before it is done. I primarily use the slope-intercept method and then x, y - intercept method.

The solution to a system of equations is the ordered pair that makes both equations true. One way to discover this point is to use a method called substitution. There are 4 very specific steps that are followed when using this method.

Procedure

1) Solve one of the equations for one of the variables (occasionally this is done for you). Usually you want to solve for the variable that is the easiest. For instance if the equation is 3x + y = 12 then you should solve for y because it does not have a coefficient. Solving for x would mean you would have to divide everything by three.

2) Look at the other equation, it should contain the variable you just solved for. The next step is to substitute for that variable what it equaled when you solved for it. This is accomplished by "replacing" the variable with the expression it equals. There should only be one variable remaining in the new equation.

3) Solve the new equation for the variable that is remaining. This will usually involve performing the distributive property first.

4) Subsitute that value into one of the two original equations and solve for the other variable.

The following examples show these four steps in action. The only thing that varies is step one. Steps 2 through 4 are pretty much the same everytime. At the end of each example the solution is checked. This is a very, very good habit to get into.

We have learned two methods for solving systems of equations. Graphing and subsitution. The goal of subsitution was to put the equations together in such a way that there was only one variable left. This allowed us to solve for the variable. The elimination method is similar. Our goal is to eliminate one of the variables by adding the two equations together. Consider the following two equations:

x + y = 12
-x +3y = 4

If we add these together vertically:

x + y = 12

+-x + 3y = 4

=0x + 4y = 16

So by adding the two equations together we eliminated the x variable. The key is that in both equations the x variable had the same coefficient (1) just an opposite sign. Thus when they are added together the coefficient of x is zero.

So with this in mind we break elimination down to the following steps.

1) The first step is to manipulate the equations such that one of the two variables will be eliminated.

2) Add the two equation together to get one equation with one variable.

3) Solve the new equation for the remaining variable.

4) Plug that value in for the variable in one of the two remaining equations and then solve for the other variable.

If you notice, steps 2-4 are very similar to steps 2-4 when using substitution. The key step is to multiply the original equations in a way that allows the variable to be eliminated when the two equations are added together.