Module 11 & 12. Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities
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1 Module 11 & 12 Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities
2 What is a System of Equations? A system of linear equations consists of two or more linear equations that have the same variables. The solution to a system of linear equations is any order pair that satisfies all of the equations in the system.
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4 Vocab: Happens when the slopes are the same, but different y intercepts. Happens when BOTH equations are exactly the same!
5 Lesson 11.1: Solving Systems by Graphing Essential Question: How can you find the solution of a system of linear equations by graphing?
6 How to Solve by Graphing 1. Graph line 1. Use the x and y intercept method Rearrange equation to slope intercept form 2. Graph line The solution is the point of intersection. 4. Write the solution as an ordered pair (x, y).
7 #1 Solve by Graphing: 2x + y = 6 x + y = 3
8 # 2 Solve by Graphing: y = 2x 2 3y + 6x =18
9 #3: Solve by Graphing: y = 2x 2 x + 2y = 2
10 #4: Solve by Graphing: y = 2x + 8 x + y = 6
11 #5: Solve by Graphing and identify solution type: y = 2x 2 2x + y = 4
12 #6: Solve by Graphing and identify solution type: y = x 2 x + y + 2= 0
13 #7: Solve by Graphing and identify solution type:
14 Rock and Bowl charges $2.75 per game plus $3 for shoe rental. Super Bowling charges $2.25 per game and $3.50 for shoe rental. For how many games will the cost to bowl be approximately the same at both places? What is the that cost?
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16 Video club A charges $10 for membership and $4 per movie rental. Video club B charges $15 for membership and $3 per movie rental. For how many movie rentals will the cost be the same at both video clubs? What is that cost? Write a system and solve by graphing.
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18 Lesson 11.2: Solving Linear Systems by Substitution Essential Question: How can you solve a system of linear equations by using substitution?
19 Theory of Substitution Method Substitution is used to reduce the system to one equation that has only one variable. Then you can solve this equation for the one variable and substitute again to find the other variable.
20 How to: Substitution Method Solving Systems of Equations by Substitution Step 1 Step 2 Step 3 Solve for one variable in at least one equation, if necessary. Substitute the resulting expression (from step 1) into the other equation. Solve that new equation to get the value of the first variable. Step 4 Step 5 Substitute the value (from step 3) into one of the original equations and solve for the other variable. Write the values from steps 3 and 4 as an ordered pair, (x, y), and check.
21 #1 Solve by Substitution: 3x + y = 3 2x + y = 7
22 #2 Solve by Substitution: x 3y = 9 x + 4y = 2
23 #3 Solve by Substitution: 3x + y = 14 2x 6y= 24
24 #4 Solve by Substitution: x + y = 4 x y= 6
25 #5 Solve by Substitution: 3x + y = 12 6x 2y= 18
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29 Lesson 11.3 & 11.4: Solving Linear Systems by Elimination Essential Question: How can you solve a system of linear equations by using multiplication and elimination?
30 How to Solve by Elimination: 1. Arrange the equations in the same form (standard form or slope intercept form). 2. Add or subtract to eliminate one variable. Simplify to solve. *****You might need to multiply to create elimination opportunities. 3. Plug in to solve for the other variable. 4. Write the answer as an ordered pair.
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32 #1 Solve Using Elimination: 4x 2y = 12 x + 2y = 8
33 #2 Solve Using Elimination: 2x + 6y = 6 2x y = 8
34 #3 Solve Using Elimination: 3x + 2y = 5 x + 2y = 1
35 #4 Solve Using Elimination: 4x 2y = 4 4x + 2y = 4
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38 #5 Solve Using Elimination: 3x + 8y = 7 2x 2y = 10
39 #6 Solve Using Elimination: 2x + 3y = 1 5x 2y = 12
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44 12.1: Creating Systems of Linear Equations Essential Question: How do you use systems of linear equations to model and solve real world problems?
45 Video streaming service Atomic Stream charges $10 for membership and $1 for each download. Blitz Video charges $15 for a membership and $.50 per movie download. How many movies would you need to download for the services to have identical costs? What is that cost?
46 Set Up the System of Equations 1. One cable television provider has a $60 setup fee and charges $80 per month. Another cable provider has a $160 equipment fee and charges $70 per month. 2. The Strauss family is deciding between two lawncare services. Green lawn charges a $49 startup fee plus $29 per month. Yard Guard charges $25 startup fee plus $37.
47 Use the cost table below to create a system of linear equations. Then solve the system to determine when the costs of the two services will be equal.
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49 Set Up the System of Equations At how many hours do both companies charge the same?
50 Use the graph to make a linear model of each function. Then create the linear system, and state what the solution represents.
51 Use the graph to make a linear model of each function. Then create the linear system, and state what the solution represents.
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54 12.2: Graphing Systems of Linear Inequalities Essential Question: How do you solve a system of linear inequalities?
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56 Solving Systems of Linear Inequalities 1. Graph Line A 2. Graph Line B 3. Shading for Line A ***Plug in (0,0).if true shade where (0,0) is if NOT true shade where (0,0) is NOT*** 4. Shading for Line B 5. The solution is where the shading overlaps.
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63 Parallel Lines No Solution: Center Solution
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65 12.3: Modeling with Linear Systems Essential Question: How can you use systems of linear equations or inequalities to model and solve contextual problems?
66 Bobby will buy coffee and hot chocolate for his coworkers. Each cup of coffee costs $2.25 and each cup of hot chocolate costs $.150. If he pays a total of for 8 cups, how many of each did he buy?
67 A student is buying pens and markers for school. Packs of pens cost $2.75 each and packs of markers cost $3.25 each. If she bought a total of 6 packs and spent $17.50 how many of each did she buy?
68 A company has to buy computers and printers. Each computer costs $550 and each printer costs $390. If the company spends $8160 and buys a total 16 machines, how many of each did it buy?
69 John has to buy two different kinds of rope. Rope A costs $0.60 per foot and Rope B costs $0.90 per foot. John needs to buy at least 15 feet of rope, but he wants to spend no more than $18. Show and describe all combinations of the number of feet of each type of rope John can buy.
70 A student has to buy graph paper and printer paper. The printer paper costs $2 a pack, while the graphing paper costs $3 a pack. She wants to buy at least 6 packs of paper but wants to spend at most $27
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Module 11 & 12. Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities
Module 11 & 12 Solving Systems of Equations Graphing Substitution Elimination Modeling Linear Systems Solving Systems of Inequalities What is a System of Equations? A system of linear equations consists
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