4.1 Systems of Equations

A system of two linear equations in two variables x and y consist of two equations of the following form:

Equation 1:

Equation 2:

where the solution (x,y) satisfies both equations.

Checking Solutions of a Linear System:

3x – 2y = 2

x + 2y = 6

1.)Is (2,2) a solution of the above system of equations?

2.)Is (0,-1) a solution of the above system of equations?

Solving a System Graphically :

Examples:

1.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator.

2x – 2y = -8

2x + 2y = 4

Check Algebraically.

.

2.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator.

3x – 2y = 6

3x – 2y = 2

3.) Solve the following system of equations graphically. Determine how many solutions. Identify the system as Consistent, Dependent Consistent, or Inconsistent. Verify your answer on your graphing calculator.

2x – 2y = -8

-2x + 2y = 8

Solving a System by Substitution

Solve each system below by the method of Substitution.

1)2)

Solve the system below by the method of Substitution, demonstrating that there is no solution.

3)

What does the graph of this system look like?

Solve the system below by the method of Substitution, demonstrating that there are infinitely many solutions.

4)

What does the graph of this system look like?

Applications

Use your graphing calculator to graph the system of equations for each application below and to answer related questions. Create a sketch of each graph, labeling the axes with appropriate scales.

1)You are checking out cell phone plans and discover that Talk Anytime Wireless charges $50.00 per month for the first phone line and charges $20.00 per additional phone line. Text Away Wireless charges $80.00 per month for the first phone line and $5.00 per additional phone line. Determine the number of additional phone lines for which it would be cheaper to use Talk Anytime verses Text Away.

2) James and Zach began saving money from their part-time jobs. James started with $50 in his savings and earns $10 per hour at his job. Zach started with $225 in his savings and earns $7.50 per hour. If both boys save all of their earnings (and we disregard tax) when will they have the same amount of savings?

3)You are choosing between two movie rental services. Company A charges $2.99 per movie plus a $20 monthly fee. Company B charges $4.99 per movie with no monthly fee. How many movies could you rent and get charged the same monthly bill? If you only rent, on average, 8 movies per month, which is the better deal for you?

Check for Understanding…

1)You are checking a solution of a system of linear equations. How can you tell if the solution is valid or not?

2)Describe how the graph of a system of linear equations looks when…

  1. There is not solution.
  1. There is exactly one solution.
  1. There are infinitely many solutions.

3)What does it mean to back-substitute when solving a system of equations?