Solving Systems of Equations Substitution Elimination (Addition)

Systems of Equations Solving systems of equations (working with two or more equations with two or more variables) allows us to find the “answer” (coordinate) that satisfies both equations. The coordinate can be found in a number of different ways: Substitution Elimination (Addition) Graphing Matrices (will not be covered in this presentation)

The System of Equations We will solve the following system of equations using the substitution, elimination (addition), and graphing methods. We will discover that the same answer can be found using the three different methods. We will discover that some equations can be solved easier using one of the three methods based on what information you are starting with.

Substitution Method Solve one of the equations for either x or y. Now that we know what y equals we can substitute that value into the second equation and solve for x. Now that we have found one of the variables, we can substitute its value back into one of the original equations: The answer, which is a coordinate, is (5, 3).

Checking Your Answer Remember to always check your answer in both equations to make sure your answer is correct. Since both of our equations are true, we can be confident that are answer of (5, 3) is correct.

Elimination Method (Addition) Using the elimination method, also known as the addition method, we want to add to the two equations together so that either the x or the y variable will drop out. If they do not drop out when added together, then we have to multiply either one or both of the equations by a number to “force” one of the variables to drop out. Remember when multiplying by a number you multiply the entire equation (both sides).

Elimination Method Add to the equations together: When we added the two equations together, the y value dropped out and solving for x was easy to do. Substitute 5 back into one of the original equations to solve for y. Coordinate to satisfy both equations is (5, 3). Don’t forget to check your answer!

Which method to use? First and foremost, follow the directions in your text book on which method to use for a particular problem. If no method is stated and you get to choose which one to use, use the method that you understand the best. If the equation has an x or y variable already with a coefficient of 1, then the substitution method works best. If the equations can be added together and one of the variables drops out, then the elimination method works best. Remember that both methods will work on the problems. You will get the same answer.

Graphing You can also graph both of the equations to find the point where the two lines cross. There are a lot of graphing programs on the Internet that work but I like to use the “Graph” program. When I graphed both of the lines, they crossed at the coordinate (5, 3).

Summary Always use the method that is given in the problem to solve a system of equations. If no method is stated, use the method that works best for you. Don’t forget to always check your answers in both equations. If they do not equal, you made a mistake somewhere and need to check your work.