The Algebra Files

Unit: Systems of Equations

Learning Targets:

Students will be able to…

  1. Identify what systems of equations are.
  2. Solve systems of equations using each of the three methods: graphing, substitution and elimination.
  3. State the solution to a system of equations as a coordinate, or explain why there are no solutions or infinitely many solutions
  4. Write equations for a system, given a word problem or a practical problem, and solve.

The Algebra Files:

A system of equation is a set of two or more equations, and the solution to a system is the point that satisfies all of the equations in that system. This means that the solution point is the point at which the equations intersect. The solution coordinate could be plugged into all of the equations, and satisfy all of them.

There are three methods of solving systems of equations:

  1. Graphing
  2. Substitution
  3. Elimination

Graphing:

Solving a system by graphing requires that you graph both equations in the system. The point where the two lines intersect is the solution point. This strategy works well when both equations are already in slope—intercept form, and both will fit on the graph provided. When graphing systems it is easy to see that parallel lines (equations that have the same slope, but a different y-intercept) will never intersect. Since they never intersect there is no solution to that system. It is also easy to see that if two equations have the same slope and the same y-intercept, they are actually the same equation, and form the same line. Since the “two” lines overlap in perpetuity, there are infinitely many solutions.

Substitution:

Solving a system using substitution requires that you solve one of the equations for one of the variables. It is most useful when at least one of the variables is already solved for, meaning that one of the equations already has the variable isolated (i.e. x = 2y + 4 already has x isolated, by itself). Once you have one of the variables isolated, you can plug what it equals into the other equation.

Elimination:

The final method used to solve systems is elimination. Elimination is best used when both equations are in standard form (meaning that they have x and y on the same side of the equation to begin with), and when the two equations already have, or can easily be made to have at least one set of coefficients that are additive inverse. When using elimination, you simply add the two equations together to eliminate one of the variables, solve for the remaining variable, and then plug your answer into the remaining equations to find the other half of the coordinate. See below.

Word Problems:

When solving word problems, you must first write the two equations given the information, and then solve!

Questions:

  1. What is a system of equations?
  1. What does it mean that the solution to a system “satisfies” both equations?
  1. What are the three methods to solving equations?
  1. Given the two equations below, what method should we use to solve? Can you solve?
  2. y = 2x + 4
  3. 3x + 5y = 7
  1. Given the two equations below, what method should we use to solve? Can you solve?
  2. 2x – y = -13
  3. 5x + y = -22