Learning Objective We will solve1 a System of two Linear Equations in two variables algebraically2. 1 find the correct answer 2 Utilizing different algebra techniques to solve equations CFU What are we going to do? What does solve mean? Solve means _________. Activate (or provide) Prior Knowledge I do You Do (WB) You Do (WB) Students, you already know what a linear equation is and how to solve for slope-intercept form. Now, we are going to Solve the System of two Linear Equations Algebraically. Write the following linear equations in Slope-Intercept Form (y = mx + b) and Identify the slope(m) and the y-intercept(b) values in each equation: 1). 6x – 3y = –3 2). 30x – 15y = –15 3). 3x + 2y = 8

Are parallel and will never cross. Concept Development A system of two linear equations is a collection of two linear equations involving the same set of variables. Solutions are the ordered pairs (x, y) that make both equations true at the same time. There are 3 types of solutions. Equations Number of Solutions One solution Infinite solutions No solutions Description If the slope-intercept form of the equations have different slopes & y-intercept, there will only be one solution. If the slope-intercept form of the equations is the same for both equations, there are infinitely many solutions. If the slope-intercept form of the equations have the same slope and different y-intercepts, there are no solutions. Graph The lines… Cross at one point. (x, y) Overlap one another. Many Solutions Are parallel and will never cross. No Solution CFU How many types of possible solutions are there for system of two linear equations and what are they? Which one of the following sets listed below is a system of two linear equations? How do you know? A. B. 3: one solution, no solution, infinite solutions In your own words, what is a system of two linear equations? A system of two linear equations is ________________________________. In your own words, what are solutions? Solutions are _______________________________________________________.

Graphing is not the only way to solve a system of equations and it is not really the best way because it has to be graphed perfectly and some answers are not integers. Since our chances of guessing the right coordinates to try for a solution are not that high, we’ll be more successful if we try a different technique: like Elimination/Addition Method. In Addition or Elimination Method, use the following five steps: Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). CFU: What 5 steps you need to follow when using Addition or Elimination Method?” Discuss with your neighbor…

Elimination/Addition Method Skill Development/Guided Practice A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve the System of two Linear Equations Algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 1a. x - y = 8 x + y = 4 2x = 12 2 2 x = 6 6 - y = 8 The solution to the system of linear equations is (6, -2). -6 -6 -y = 2 y = -2 CFU (#1a) How did I/you multiply an equation to set up elimination of variables? (#2) How did I/you eliminate one of the variables? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable? CFU (#1) How did I/you isolate one of the variables? (#2) How did I/you substitute the value of the variable? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable?

Elimination/Addition Method Skill Development/Guided Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 2a. x + y = 8 x – y = -4 2x = 4 2 2 x = 2 2 + y = 8 The solution to the system of linear equations is (2, 6). -2 -2 y = 6 CFU (#1a) How did I/you multiply an equation to set up elimination of variables? (#2) How did I/you eliminate one of the variables? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable? CFU (#1) How did I/you isolate one of the variables? (#2) How did I/you substitute the value of the variable? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable?

Elimination/Addition Method Skill Development/Guided Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve the System of two Linear Equations Algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 3a. 2x + 2y = 8 2x + 2y = 8 -2x – 2y = -8 -2 ( ) x + y = 4 0 = 0 There are infinite solutions. CFU (#1a) How did I/you multiply an equation to set up elimination of variables? (#2) How did I/you eliminate one of the variables? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable? CFU (#1) How did I/you isolate one of the variables? (#2) How did I/you substitute the value of the variable? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable?

Elimination/Addition Method Skill Development/Guided Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 4a. 2x – 2y = 10 2x – 2y = 10 -2x + 2y = -10 -2 ( ) x – y = 5 0 = 0 There are infinite solutions. CFU (#1a) How did I/you multiply an equation to set up elimination of variables? (#2) How did I/you eliminate one of the variables? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable? CFU (#1) How did I/you isolate one of the variables? (#2) How did I/you substitute the value of the variable? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable?

Elimination/Addition Method Skill Development/Guided Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve the System of two Linear Equations Algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 5a. -2x – 4y = -12 -2 ( ) x + 2y = 6 2x + 4y = 20 2x + 4y = 20 0 ≠ 8 There are no solutions. CFU (#1a) How did I/you multiply an equation to set up elimination of variables? (#2) How did I/you eliminate one of the variables? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable? CFU (#1) How did I/you isolate one of the variables? (#2) How did I/you substitute the value of the variable? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable?

Elimination/Addition Method Skill Development/Guided Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 6a. -2x – 4y = -16 -2 ( ) x + 2y = 8 2x + 4y = 12 2x + 4y = 12 0 ≠ -4 There are no solutions. CFU (#1a) How did I/you multiply an equation to set up elimination of variables? (#2) How did I/you eliminate one of the variables? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable? CFU (#1) How did I/you isolate one of the variables? (#2) How did I/you substitute the value of the variable? (#3) How did I/you solve for the first variable? (#4) How did I/you solve for the second variable?

Relevance A system of two linear equations is a collection of two linear equations involving the same set of variables. 1. Solving a system of two linear equations will help you in real-life situations, such as keeping track of money. 2. Solving a system of two linear equations will help you do well on tests. Below, there are 20 total bills and the value of all the money is $235. How many $5 bills are there? How many $20 bills are there? # of $5 bills # of $20 bills Solve this system of equations to answer the questions. CFU Does anyone else have another reason why it is relevant to solve a system of two linear equations? (Pair-Share) Why is it relevant to solve a system of two linear equations? You may give me one of my reasons or one of your own. Which reason is most relevant to you? Why?

Elimination/Addition Method Skill Development/Guided Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Understand the problem. Read and reread the problem, Choose a variable to represent the unknown, and Construct a drawing, whenever possible Step #2: Translate the problem into two equations. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 1a. Delhi High School club sold 311 tickets for a play. Student tickets cost 50 cents each; non student tickets cost $1.50. If total receipts were $385.50, find how many tickets of each type were sold. 1.) Understand s = the number of student tickets n = the number of non-student tickets 5.) Interpret State: There were 81 student tickets and 230 non student tickets sold. 2.) Translate DHS club sold 311 tickets for a play.  s + n = 311 Admission for students + Admission for non students = Total receipts  0.50s + 1.50n = 385.50 4.) Sub-in 3.) Solve s + n = 311 s – 3n = 771 s + n = 311 s + n = 311 2(0.50s + 1.50n) = 2(385.50) 2n = 460 s + 230 = 311 n = 230 s = 81

Elimination/Addition Method Skill Development/Guided Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Understand the problem. Read and reread the problem, Choose a variable to represent the unknown, and Construct a drawing, whenever possible Step #2: Translate the problem into two equations. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 2a. Delhi High School club sold 400 tickets for a play. Student tickets cost 75 cents each; non student tickets cost $3. If total receipts were $450, find how many tickets of each type were sold. 1.) Understand s = the number of student tickets n = the number of non-student tickets 5.) Interpret State: 2.) Translate 3.) Solve 4.) Sub-in

Elimination/Addition Method A system of two linear equations is a collection of two linear equations involving the same set of variables. Skill Closure Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method x + y = 6 x – y = 2 2x = 8 2 2 x = 4 The solution to the system of linear equations is (4, 2). 4 + y = 6 -4 -4 y = 2 Constructed Response Closure Describe the graphs of a system of two linear equations where there is a single solution, infinite solutions, and no solution. Summary Closure What did you learn today about solving a system of two linear equations? (Pair-Share) Day 1 ___________________________________________________________________________________________________ Day 2 ___________________________________________________________________________________________________

Elimination/Addition Method Independent Practice Name _________________ A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 1a. 2y + x = 5 -2y + x = 1 2x = 6 2 2 x = 3 2y + 3 = 5 The solution to the system of linear equations is (3, 1). -3 -3 2y = 2 2 2 y = 1

Elimination/Addition Method Independent Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 2a. -3x – 6y = -36 -3 ( ) x + 2y = 12 3x + 6y = 36 3x + 6y = 36 0 = 0 There are infinite solutions.

Elimination/Addition Method Independent Practice (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 3a. 3x + 9y = 18 3x + 9y = 18 -3x - 9y = -27 -3 ( ) x + 3y = 9 0 ≠ -9 There are no solutions.

Elimination/Addition Method Periodic Review 1 Name _________________ A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 1a. x - y = 8 x + y = 4 2x = 12 2 2 x = 6 6 - y = 8 The solution to the system of linear equations is (6, -2). -6 -6 -y = 2 y = -2

Elimination/Addition Method Periodic Review 1 (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 2a. x + 2y = 12 x – 2y = 4 2x = 16 2 2 x = 8 8 + 2y = 12 The solution to the system of linear equations is (8, 2). -8 -8 2y = 4 y = 2

Elimination/Addition Method Periodic Review 1 (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 3a. -8x – 8y = -16 -8 ( ) x + y = 2 8x + 8y = 16 8x + 8y = 16 0 = 0 There are infinite solutions.

Elimination/Addition Method Periodic Review 2 Name _________________ A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 1a. x - y = 6 x + y = 2 2x = 8 2 2 x = 4 4 - y = 6 The solution to the system of linear equations is (4, -2). -4 -4 -y = 2 y = -2

Elimination/Addition Method Periodic Review 2 (continued) A system of two linear equations is a collection of two linear equations involving the same set of variables. Solve a system of two linear equations in two variables algebraically. Step #1: Line up the two equations in standard form, one on top of the other. a: If necessary, multiply an equation by a number to set up an elimination of variables. Step #2: Add the two equations to eliminate one of the variables. Step #3: Solve for the first variable. Step #4: Substitute to solve for the second variable. Step #5: Interpret your answer as order pair (x, y). Elimination/Addition Method 2a. 3( ) -x + 2y = 2 -3x + 6y = 6 3x – 6y = 9 3x – 6y = 9 0 ≠ 15 There are no solutions.