 SOLVE A SYSTEM OF TWO LINEAR EQUATIONS IN TWO VARIABLES BY GRAPHING.  SOLVE A SYSTEM OF TWO LINEAR EQUATIONS IN TWO VARIABLES USING THE SUBSTITUTION AND THE ELIMINATION METHODS.  USE SYSTEMS OF TWO LINEAR EQUATIONS TO SOLVE APPLIED PROBLEMS. Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley 9.1 Systems of Equations in Two Variables

Systems of Equations Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley A system of equations is composed of two or more equations considered simultaneously. Example: 5x  y = 5 4x  y = 3 This is a system of two linear equations in two variables. The solution set of this system consists of all ordered pairs that make both equations true. The ordered pair (2, 5) is a solution of this system.

Solving Systems of Equations Graphically Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley When we graph a system of linear equations, each point at which the graphs intersect is a solution of both equations and therefore a solution of the system of equations.

Solving Systems of Equations Graphically Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Let’s solve the previous system graphically. 5x  y = 5 4x  y = 3

Types of Solutions Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Graphs of linear equations may be related to each other in one of three ways.

Practice Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Method 1 – Substitution Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley The substitution method is a technique that gives accurate results when solving systems of equations. It is most often used when a variable is alone on one side of an equation or when it is easy to solve for a variable. One equation is used to express one variable in terms of the other, then it is substituted in the other equation.

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Use substitution to solve the system 5x  y = 5, 4x  y = 3.

Method 2 – Elimination Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Using the elimination method, we eliminate one variable by adding the two equations. If the coefficients of a variable are opposites, that variable can be eliminated by simply adding the original equations. If the coefficients are not opposites, it is necessary to multiply one or both equations by suitable constants, before we add.

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve the system using the elimination method. 6x + 2y = 4 10x + 7y =  8

Another Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve the system. x  3y =  9 2x  6y = 3

Another Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Solve the system. 9x + 6y = 48 3x + 2y = 16

Substitution Elimination Practice Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley

Application--Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley Ethan and Ian are twins. They have decided to save all of the money they earn, at their part-time jobs, to buy a car to share at college. One week, Ethan worked 8 hours and Ian worked 14 hours. Together they saved $256. The next week, Ethan worked 12 hours and Ian worked 16 hours and they earned $324. How much does each twin make per hour?

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley At Max’s Munchies, caramel corn worth $2.50 per pound is mixed with honey roasted mixed nuts worth $7.50 per pound in order to get 20 lb. of a mixture worth $4.50 per pound. How many of each snack is used? Carmel corn NutsMixture Price per pound $2.50$7.50$4.50 Number of pounds xy20 Value of Mixture 2.50x7.50y4.50(20) = 90

Example Copyright © 2012 Pearson Education, Inc. Publishing as Addison Wesley An airplane flies the 3000-mi distance from Los Angeles to New York, with a tailwind, in 5 hr. The return trip, against the wind, takes 6 hr. Find the speed of the airplane and the speed of the wind. DistanceRateTime With Tailwind 3000p + w5 With headwind 3000p – w6