Algebra 1 End-of-Course Review

Module Six: Systems of Equations

In this module you will review three ways to solve systems of equations and inequalities, and also take a look at some word problems.

Solving Systems of Equations

  • A system of equations is a collection of two or more equations.
  • The solution to a system of equations (in two variables) is the pair of values that make both equations true. This pair of values corresponds to the point where the two lines intersect when graphed on a coordinate plane.
  • There are three ways to solve a system of equations:
    1. Graphically: graphing both equations and finding the point of intersection

    2. Algebraically:
    • Substitution Method – isolating one variable in one of the equations and
      substituting it into the other equation

    • Elimination Method – eliminating one of the variables when combining the
      two equations

Remember that each method will result in the same solution. To demonstrate this, solve the following system of equations using all three methods.

x – 2y = 2
3x + y = 6

Special Cases

There are three types of solutions of systems of equations. They are shown here as Case 1, Case 2, and Case 3. Select the graphs of Case 2 and Case 3 to show how these two special cases are solved.

Case 1 Case 2 Case 3
Case 1 is a graph of two infinite lines, one with a positive slope and one with a negative slope that intersect at a point. Case 2 is a graph of two infinite parallel lines, both with a positive slope. Case three is a graph of one line with a positive slope
Intersecting lines Parallel lines Same line
One Solution – the point of intersection No solution – the lines never intersect Infinitely Many Solutions – the two equations graph the same line
Algebraically one solution for each variable Algebraically the variables cancel and result in a false statement Algebraically the variables cancel and result in a true statement

Applications of Systems

Now that you know how to solve systems of equations, you can apply that knowledge to writing and solving systems of equations for distance, age, and money problems. An example of each type is reviewed here.

Look here for examples of money, distance, and age problems.

 

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