Slide 1- 1 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Systems of Equations in Two Variables Translating Identifying Solutions Solving Systems Graphically 8.1

Slide 3- 3 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley System of Equations A system of equations is a set of two or more equations, in two or more variables, for which a common solution is sought.

Slide 3- 4 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each. In all, $ was taken in for the shirts. How many of each kind were sold? Set up the equations but do not solve. Example

Slide 3- 5 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Identifying Solutions A solution of a system of two equations in two variables is an ordered pair of numbers that makes both equations true.

Slide 3- 6 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Determine whether (1, 5) is a solution of the system Example

Slide 3- 7 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solving Systems Graphically One way to solve a system of two equations is to graph both equations and identify any points of intersection. The coordinates of each point of intersection represent a solution of that system.

Slide 3- 8 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve the system graphically. Example x – y = 1 x + y = 5 (3, 2)

Slide 3- 9 Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve the system graphically. Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve the system graphically. Solution Example Graph both equations. The same line is drawn twice. Any solution of one equation is a solution of the other. There is an infinite number of solutions. The solution set is

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When we graph a system of two linear equations in two variables, one of the following three outcomes will occur. 1.The lines have one point in common, and that point is the only solution of the system. Any system that has at least one solution is said to be consistent. 2.The lines are parallel, with no point in common, and the system has no solution. This type of system is called inconsistent. 3.The lines coincide, sharing the same graph. This type of system has an infinite number of solutions and is also said to be consistent.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley When one equation in a system can be obtained by multiplying both sides of another equation by a constant, the two equations are said to be dependent. If two equations are not dependent, they are said to be independent.

Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Solving by Substitution or Elimination The Substitution Method The Elimination Method 8.2

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Substitution Method Algebraic (nongraphical) methods for solving systems are often superior to graphing, especially when fractions are involved. One algebraic method, the substitution method, relies on having a variable isolated.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Solve the system(1) (2) The equations are numbered for reference. Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Solve the system(1) (2) Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Solve the system (1) (2) Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley The Elimination Method The elimination method for solving systems of equations makes use of the addition principle: If a = b, then a + c = b + c.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve the system (1) (2) Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solve the system (1) (2) Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Solve the system (1) (2) Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Rules for Special Cases When solving a system of two linear equations in two variables: 1. If we obtain an identity such as 0 = 0, then the system has an infinite number of solutions. The equations are dependent and, since a solution exists, the system is consistent. 2. If we obtain a contradiction such as 0 = 7, then the system has no solution. The system is inconsistent.

Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison- Wesley Solving Applications: Systems of Two Equations 8.3

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Total-Value Problems The next example involves two types of items, the quantity of each type bought, and the total value of the items. We refer to this type of problem as a total-value problem.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley T-shirt Villa sold 52 shirts, one kind at $8.25 and another at $11.50 each. In all, $ was taken in for the shirts. How many of each kind were sold? Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Mixture Problems The next example is similar to the last example. Note that in each case, one of the equations in the system is a simple sum while the other equation represents a sum of products. We refer to this type of problem as a mixture problem.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Problem-Solving Tip When solving a problem, see if it is patterned or modeled after a problem that you have already solved.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution An employee at a small cleaning company wishes to mix a cleaner that is 30% acid and another cleaner that is 50% acid. How many liters of each should be mixed to get 20 L of a solution that is 35% acid? + 30% acid 50% acid 35% acid = t liters f liters 20 liters Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Motion Problems The next example deals with distance, speed (rate), and time. We refer to this type of problem as a motion problem.

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Distance, Rate, and Time Equations If r represents rate, t represents time, and d represents distance, then

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Alex’s motorboat took 4 hr to make a trip downstream with a 5-mph current. The return trip against the same current took 6 hr. Find the speed of the boat in still water. Example

Slide Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Tips for Solving Motion Problems 1.Draw a diagram using an arrow or arrows to represent distance and the direction of each object in motion. 2.Organize the information in a chart. 3.Look for times, distances, or rates that are the same. These often can lead to an equation. 4.Translating to a system of equations allows for the use of two variables. 5.Always make sure that you have answered the question asked.