Chapter 7: Systems of Linear Equations Section 7.1 Chapter 7: Systems of Linear Equations Section 7.1: Developing Systems of Linear Equations Terminology: System of Linear Equations: A grouping of two or more equations involving the same variables. For the purposes of this course we will only be working with two variable systems or as they are more commonly referred to Linear Systems. Setting Up A Linear System When given a situation, you will be expected to be able to set up a correct system of equations. Steps: 1. State the variables you will be using (ex. C = cost, t = time, n = necklaces). 2. Read the question carefully and pick out the two different statements that will represent out two equations in our system. 3. Write both of the equations using the stated variables. Example: Creating a System of Equations (a) Internet Company A charges a monthly base fee of $20 and an additional 10 per GB used. Whereas Company B charges 5 per GB and $30 per month. Create a system of equations to represent this situation. 137
Chapter 7: Systems of Linear Equations Section 7.1 (b) A local movie theatre sells tickets at $5.50 for children and $7.75 for adults. During the premiere of the new Avengers movie they made $4526.25. If 645 people attended the premiere, set up an appropriate system of equations. (c) Emily bought 8 books at Chapters last weekend. Some books cost $13 each and the rest cost $24 each. She spent a total of $209. Write a system of equations that could represent this situation. (d) A Theatre charges $4 for children and $7 for adults. During this weekend s premier, 40 people attended a movie and the Theatre made a profit of $256. Write a system of equations that could represent this situation. 138
Chapter 7: Systems of Linear Equations Section 7.2 Section 7.2: Solving a System using Graphing Yes, you guessed it. To solve a system of equations graphically we must indeed be graphing some lines. Get you y = mx + b skills ready because we gonna use em!! All that we need to do is graph two lines and determine where they intersect Example 1: Solving A Linear System By Graphing Solve this linear system: x + y = 8 { 3x 2y = 14 10 8 6 4 2 y - 10-8 - 6-4 - 2 2 4 6 8 10-2 x - 4-6 - 8-10 Example 2: Solve this system of Equations x + y = 2 { x 2y = 2 10 8 6 4 2 y - 10-8 - 6-4 - 2 2 4 6 8 10-2 x - 4-6 - 8-10 139
Chapter 7: Systems of Linear Equations Section 7.2 Example 3: Solving a Problem by Creating the System then Graphing (a) Write the linear system to model this situation: To visit the Head-Smashed-In in Buffalo Jump Interpretative Centre near Fort Macleod, Alberta, the admission fee is $5 for students and $9 for adults. In one hour, 32 people entered the centre and a total of $180 in admission fees was collected. (b) Graph the linear system then solve this problem: How many students and how many adults visited the centre during this hour? y x 140
Chapter 7: Systems of Linear Equations Section 7.2 Example 4: (a) Write the linear system to model this situation: Wayne received and sent 60 text messages on his cell phone in one weekend. He sent 10 more messages than he received. (b) Graph the linear system and solve this problem: How many text messages did Wayne send and how many did he receive? y x 141
Chapter 7: Systems of Linear Equations Section 7.4 Section 7.4: Solving Systems using Substitution Solving a System of Linear Equations by SUBSTITUTION Steps: 1. Rearrange one of the equations for a single variable. Note: When selecting a variable to solve for, look for variables with easily manageable coefficients, such as 1, or ones with common factors throughout the equation. 2. Substitute the solved variable from step 1 into the second equation in the system. 3. Solve the second equation for the remaining variable. 4. Substitute the result of step 3 back into the equation from step 1. 5. State you solution. Example 1: Solve the system by substitution 3x + 4y = 4 (a) { x + 2y = 2 NOTE: To verify a solution to be correct, plug it back into each equation in the system to see if they work out to the correct values. 142
Chapter 7: Systems of Linear Equations Section 7.4 2x 4y = 7 (b) { 4x + y = 5 (c) { 3x 2y = 13 2x + 5y = 53 143
Chapter 7: Systems of Linear Equations Section 7.4 Example 2: Using Substitution to Solve Word Problems (a) Nuri invested $2000, part of it at an annual interest rate of 8% and the rest at an annual interest rate of 10%. After one year, the total interest earned was $190. Use substitution to determine how much Nuri invested at each rate. (b) Tamara bought 17 books at Chapters last weekend. Some books cost $13 each and the rest cost $24 each. She spent a total of $309. Us substitution to determine how many of each type of book she bought. (c) A Theatre charges $4 for children and $7 for adults. During this weekend s premier, 40 people attended a movie and the Theatre made a profit of $256. Write a system of equations that could represent this situation. 144
Chapter 7: Systems of Linear Equations Section 7.5 Section 7.5: Solving Systems using Elimination Solving a System of Linear Equations by ELIMINATION Steps: 1. Multiply one or both equations by an integer number to make one variable in each equation have the same coefficient (if necessary). 2. Add or subtract the two equations (whatever operation is required) to cancel out (ELIMINATE) the variable from step 1. 3. Solve the system for the remaining variable. 4. Substitute the solution from step 3 into one of the original equations and solve for the remaining variable 5. State you solution. Example 1: Solve the system via elimination 2x + y = 7 (a) { x + y = 4 NOTE: To verify a solution to be correct, plug it back into each equation in the system to see if they work out to the correct values. Just as we did before for substitution. 145
Chapter 7: Systems of Linear Equations Section 7.5 3x 4y = 7 (b) { 5x 6y = 8 (c) { 2x + 7y = 24 3x 2y = 4 146
Chapter 7: Systems of Linear Equations Section 7.5 Alternatively we can also solve a system by repeating the first three steps of the process for each variable Example 2: Solve using Elimination 2x + 3y = 8 { 5x 4y = 6 Example 3: Using Elimination for Word Problems (a) A local movie theatre sells tickets at $5.50 for children and $7.75 for adults. During the premiere of the new Avengers movie they made $4526.25. If 645 people attended the premiere, determine the number of children and adults in attendance. 147
Chapter 7: Systems of Linear Equations Section 7.5 (b) A bake sale charges $3.25 for a slice of cake and $4.75 for a tray of cookies. If the bake sale sold 108 items and made $477.00, determine how many pieces of cake and how many trays of cookies were sold. 148
Chapter 7: Systems of Linear Equations Section 7.6 Section 7.6: Systems of Equations Special Cases y It is important to note that all the systems that we have looked at thus far have had only one solution (ie. Two lines will cross at exactly one point). There are two other possibilities when solving a system: 10 5-10 - 5 5 10 x - 5-10 1. The system has no possible solutions: - This occurs when the two equations that we are working with correspond to two parallel lines. - This can be identified when solving a system by substitution or elimination if all variables cancel in the process leaving behind only a false statement (ex. 5 = 0) y 10 5-10 - 5 5 10-5 - 10 x 2. The system has infinite solutions: - This occurs when the two equations we are working with correspond to the same line (ie. The lines are superimposed on one another). - This can be identified when solving a system by substitution or elimination if all variables cancel in the process leaving behind only a true statement (ex. 0 = 0) y 10 5-10 - 5 5 10-5 x - 10 149
Chapter 7: Systems of Linear Equations Section 7.6 Ex. Solve each of the following solutions and determine if there are no solutions, one solution, or infinitely many solutions. 6x + 9y = 3 (a) { 4x + 6y = 2 x 3y = 8 (b) { 4x + 2y = 18 (c) 4x 10y = 8 { 10x 25y = 15 150