1.2. Monetary Systems Overload. Systems of Linear Equations. ACTIVITY 1.2 Guided

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1 Sstems of Linear Equations SUGGESTED LEARNING STRATEGIES: Shared Reading, Close Reading, Interactive Word Wall Have ou ever noticed that when an item is popular and man people want to bu it, the price goes up? Have ou ever noticed that items that no one wants are marked down to a lower price? The change in an item s price and the quantit available to bu are the basis of the concept of suppl and demand in economics. Demand refers to the quantit that people are willing to bu at a particular price. Suppl refers to the quantit that the manufacturer is willing to produce at a particular price. The final price that the customer sees is a result of both suppl and demand. Suppose that during a si-month time period, the suppl and demand for gasoline has been tracked and approimated b these functions, where Q represents millions of barrels of gasoline and P represents price per gallon in dollars. Demand function: P = -0.7Q Suppl function: P = 1.5Q To find the best balance between market price and quantit of gasoline supplied, find a solution of a sstem of two linear equations. The demand and suppl functions for gasoline are graphed below p ACTIVITY 1.2 A point, or set of points, is a solution of a sstem of equations in two variables when the coordinates of the points make both equations true. Guided Sstems of Linear Equations Activit Focus Solving sstems of equations in two variables b graphing Solving sstems of equations in two variables b substitution Solving sstems of equations in two and three variables b elimination Classifing sstems of equations in two or three variables Materials Whiteboard or chart paper Grid paper (optional) Chunking the Activit #1 2 #5 #3 4 #6 Price (dollars) Introduction Shared Reading, Close Reading, Interactive Word Wall 12 The first few items will introduce solving sstems of linear equations b graphing. Item 1 also demonstrates the limitations of graphing as a solution method. It asks students to approimate the solution b identifing a point of intersection that is not a lattice point in the coordinate plane Gasoline (millions of barrels) 1. Find an approimation of the coordinates of the intersection of the suppl and demand functions. Eplain what the point represents. Answers ma var. Sample answer: (9.15, 3.3); At a price of $3.30, people will demand 9.15 million gallons of gas, and companies will be willing to suppl it. Q Unit 1 Linear Sstems and Matrices 13 Unit 1 Linear Sstems and Matrices 13

2 Continued 3 Create Representations Remind students that to graph an equation, the should either write the equation in slopeintercept form or find the - and -intercepts. Technolog Tip Students can use graphing calculators to graph each sstem and determine its solution. See Mini-Lesson: Solving Sstems Using a Graphing Calculator below. Sstems of Linear Equations TECHNOLOGY You can use a graphing calculator and its TRACE function to solve sstems of equations in two variables. SUGGESTED LEARNING STRATEGIES: Create Representations, Vocabular Organizer 2. What problem(s) can arise when solving a sstem of equations b graphing? Answers ma var. Sample answer: Graphing is not ver accurate if the intersection is not on a lattice point, or the scaling of the graph is not accurate enough. 3. Graph each sstem. Determine the number of solutions. a. { = + 1 = + 4 one solution Vocabular Organizer Have students share their graphs and their sstem classification choices. If necessar, have students graph a few more equations to reinforce the characteristics of different sstem classifications. Using a graphing calculator can speed this process. Sstems of linear equations are classified b the number of solutions. Sstems with no solution are inconsistent. Sstems with one or man solutions are consistent. A sstem with eactl one solution is independent. A sstem with infinite solutions is dependent. b. { = = 2 no solution c. { = = infinitel man solutions Graphing two linear equations illustrates the relationships of the lines. Classif the sstems in Item 3 as consistent and independent, consistent and dependent, or inconsistent. a. independent and consistent b. inconsistent c. dependent and consistent 14 SpringBoard Mathematics with Meaning Algebra 2 MINI-LESSON: Solving Sstems Using a Graphing Calculator Press Y =. In the Y1 row, tpe the first equation. In Y2, tpe the second equation. Press GRAPH. Be sure that our range makes the intersection visible. Press 2nd and TRACE. Press 5 to select 5:intersection. Move the cursor near the intersection point and press ENTER twice to select the curves. Press ENTER again for the Guess? 14 SpringBoard Mathematics with Meaning Algebra 2

3 Sstems of Linear Equations Continued SUGGESTED LEARNING STRATEGIES: Note Taking, Look for a Pattern, Think/Pair/Share Investors tr to control the level of risk in their portfolios b diversifing their investments. You can solve some investment problems b writing and solving sstems of equations. One algebraic method for solving a sstem of linear equations is called substitution. EXAMPLE 1 Note Taking Walk students through the eample. Some students ma find it easier to work with whole numbers. Have them multipl the second equation b 100 to rewrite it as = 20,500. EXAMPLE 1 During one ear, Sara invested $5000 into two separate funds, one earning 2% and another earning 5% annual interest. The interest Sara earned was $205. How much mone did she invest in each fund? Step 1: Step 2: Let = mone in the first fund and = mone in the second fund. Write one equation to represent the amount of mone invested. Write another equation to represent the interest earned. TRY THESE A Use these eercises to determine which students need etra help with the process. Use questioning strategies to help students solidif their understanding. Some students ma benefit from listing each step as the work through the solutions. 5 Think/Pair/Share, Look for a Pattern Have volunteers share their answers to this question. Focus a discussion on wh it is helpful to look for a variable with a coefficient of 1 first, and then, if there are no such variables, to look for a variable with a coefficient of -1 net. Suggested Assignment CHECK YOUR UNDERSTANDING p. 20, #1 2 UNIT 1 PRACTICE p. 69, #7 8 + = 5000 The mone invested is $ = 205 The interest earned is $205. Use substitution to solve this sstem. + = 5000 Solve the first equation for. = ( ) = 205 Substitute for in the second equation = 205 Solve for = -45 = 1500 Step 3: Substitute the value of into one of the original equations to find. + = = 5000 Substitute 1500 for. = 3500 Solution: Sara invested $1500 in the first fund and $3500 in the second fund. TRY THESE A Write our answers on notebook paper. Show our work. Solve each sstem of equations, using substitution. Check students' work. = 25 3 a. { = 9 b. { + 2 = 14 c. 2 = - 10 { = = 16 (-14, 13) (12, 1) (3, 7) In the substitution method, ou solve one equation for one variable in terms of another. Then substitute that epression into the other equation to form a new equation with onl one variable. Solve that equation. Substitute the solution into one of the two original equations to find the other variable. Check our answer b substituting the solution (1500, 3500) into the second original equation = When using substitution, how do ou decide which variable to isolate and which equation to solve? Eplain. Answers will var. Sample answer: Choose a variable that is eas to isolate b finding the equation with a variable that has a coefficient of 1 or -1. Unit 1 Linear Sstems and Matrices 15 Unit 1 Linear Sstems and Matrices 15

4 Continued EXAMPLE 2 Note Taking Work through the eample with students. Refer to the Math Terms bo for a summar of how to use the elimination method. Point out the importance of multipling both sides of one equation b a number that will allow one variable term to be eliminated when the equations are added. TEACHER TO Students ma question TEACHER wh the have to learn more than one wa to solve a sstem of equations. Allow students to compare and contrast the methods b having them solve one or more of the following sstems using each method = =-16 (-3, 4) = = 5 (-5, 3) = = 11 (2, -1) Sstems of Linear Equations The elimination method is also called the addition-elimination or the linear combination method for solving a sstem of linear equations. In the elimination method, ou eliminate one variable. Multipl each equation b a number so that the terms for one variable combine to 0 when the equations are added. Then use substitution with that value of the variable to find the value of the other variable. The ordered pair is the solution of the sstem. SUGGESTED LEARNING STRATEGIES: Note Taking Another algebraic method for solving sstems of linear equations is the elimination method. EXAMPLE 2 A stack of 20 coins contains onl nickels and quarters and has a total value of $4. How man of each coin are in the stack? Step 1: Let n = number of nickels and q = number of quarters. Write one equation to represent the number of coins in the stack. Write another equation to represent the total value. Step 2: Step 3: n + q = 20 The number of coins is 20. 5n + 25q = 400 The total value is 400 cents. To solve this sstem of equations, first eliminate the n variable. -5(n + q) = -5(20) Multipl the first equation b -5. 5n + 25q = 400-5n - 5q = n + 25q = 400 Add the two equations to eliminate n. 20q = 300 Solve for q. q = 15 Find the value of the eliminated variable n b using the original first equation. n + q = 20 n + 15 = 20 Substitute 15 for q. n = 5 Step 4: Check our answers b substituting into the original second equation. 5n + 25q = 400 5(5) + 25(15)? 400 Substitute 5 for n and 15 for q ? = 400 Check. Solution: There are 5 nickels and 15 quarters in the stack of coins. 16 SpringBoard Mathematics with Meaning Algebra 2 16 SpringBoard Mathematics with Meaning Algebra 2

5 Sstems of Linear Equations Continued SUGGESTED LEARNING STRATEGIES: Close Reading, Vocabular Organizer TRY THESE B Solve each sstem of equations, using elimination. Write our answers in the space. Show our work. Check students work. 2 3 = 5 a. { = 40 b. { = 14 c. 2 = 10 { = 21 5 = 17 (5, -5) (2, -4) (-3, 4) TRY THESE B Use these questions as a formative assessment. If students need more scaffolding, have them use the section to write out each step. Remind students to check their results b substituting into one of the original equations. Sometimes a situation has more than two pieces of information. For these more comple problems, ou ma need to solve equations that contain three variables. In Bisbee, Arizona, an old mining town, ou can bu souvenir nuggets of gold, silver, and bronze. For $20, ou can bu an of these mitures of nuggets: 14 gold, 20 silver, 24 bronze; or, 20 gold, 15 silver, 19 bronze; or, 30 gold, 5 silver, 13 bronze. What is the monetar value of each souvenir nugget? The problem above represents a sstem of linear equations in three variables. The sstem can be represented with these equations. Suggested Assignment CHECK YOUR UNDERSTANDING p. 20, #3 UNIT 1 PRACTICE p. 69, #9, 11 14g + 20s + 24b = 20 20g + 15s + 19b = 20 30g + 5s + 13b = 20 Although it is possible to solve sstems of equations in three variables with the substitution method, it can be difficult. It can also be ver challenging to solve this kind of sstem b graphing. Just as the ordered pair (, ) is a solution of a sstem in two variables, the ordered triple (,, z) is a solution of a sstem in three variables. Ordered triples are graphed in three-dimensional coordinate space. The point (3, -2, 4) is graphed below. z An ordered pair can also be the solution of a single equation in two variables. Likewise, an ordered triple can also be the solution of a single equation in three variables. Introduction Close Reading, Vocabular Organizer Solving this contetual problem is not part of this activit. The problem is used simpl to illustrate one tpe of situation that can be represented b three variables in a sstem of linear equations. (3, 2, 4) 4 units up O 2 units left 3 units forward Unit 1 Linear Sstems and Matrices 17 Unit 1 Linear Sstems and Matrices 17

6 Continued EXAMPLE 3 Note Taking Be sure to point out that solving sstems in three variables is similar to solving sstems in two variables once a variable term has been eliminated. In this case, after the -terms are eliminated from the three original equations, students will be solving a sstem in two variables: and z. Differentiating Instruction For students who need a challenge beond Eample 3, assign the problem of finding the monetar value of the souvenir nuggets on the previous page. The correct solution is g = $0.50, s = $0.35, b = $0.25. Sstems of Linear Equations You can use either elimination or substitution to solve a sstem of three equations in three variables. Use elimination if the terms easil add to 0. Use substitution if one equation has onl one variable on one side, such as = 2 + z. SUGGESTED LEARNING STRATEGIES: Note Taking The elimination method is usuall the easiest method for solving sstems of equations in three variables. EXAMPLE z = -53 Solve this sstem, using elimination z = z = -45 Step 1: Add the first and second equations to eliminate. Step 2: z = z = z = -66 Use the second and third equations to eliminate again. Multipl the second equation b 3 so that the -terms add to zero. 3( z) = 3(-13) z = z = z = z = -84 Step 3: Use the two equations ou found to write a new sstem in two variables. Multipl the second equation b -2 so that the z-terms add to zero. New sstem Multipl the first Add the equations equation b -2. to eliminate z. As a final step, check our ordered triple solution in another original equation to be sure that our solution is correct z = z = (10 + 2z) = -2(-66) z = - 84 Solve for. Step 4: Substitute the -value into one of the two new equations to find z z = (-6) + 2z = z = -66 2z = -6 z = -3 Step 5: Substitute the - and z-values into one of the original equations to find z = (-6) + -3 = = = -4 Solution: The solution of the sstem is (-4, - 6, -3) z = z = = 48 = SpringBoard Mathematics with Meaning Algebra 2 18 SpringBoard Mathematics with Meaning Algebra 2

7 Sstems of Linear Equations Continued SUGGESTED LEARNING STRATEGIES: Note Taking, Vocabular Organizer, Look for a Pattern TRY THESE C Solve each sstem of equations using elimination. Write our answers on notebook paper. Show our work. Check students work. a. + + z = z = z = 9 b. (1, 2, 3) (2, 4, 1) z = z = z = 5 The graph of an equation in three variables is a plane in coordinate space. z TRY THESE C Use these questions to clarif an questions or misunderstandings about the elimination method that students ma still have. It ma be helpful to have students work in groups, with each group choosing a different variable to eliminate. Then have the groups share their work. This will demonstrate that there are multiple was of finding the solution to a sstem of three equations. Paragraph Note-Taking You can represent a sstem of equations in three variables as three planes in the same coordinate space. The graph of the solutions of a sstem in three variables is the intersection of the three planes. Onl those points that form the intersection of all three planes represent the solution. 6. Classif each sstem as consistent and independent, consistent and dependent, or inconsistent. a. b. O 6 Vocabular Organizer, Look for a Pattern Students will replicate what the did earlier in two dimensions, but now with three dimensions. The vocabular organizer used earlier in Item 4 can be etended to include the three-dimensional graphs. Sstems of linear equations in three variables can be classified in the same wa as sstems in two variables. See Math Terms on page 14. independent and consistent independent and inconsistent Unit 1 Linear Sstems and Matrices 19 Unit 1 Linear Sstems and Matrices 19

8 Continued 6 Look for a Pattern Sstems of Linear Equations SUGGESTED LEARNING STRATEGIES: Look for a Pattern Suggested Assignment CHECK YOUR UNDERSTANDING p. 20, #4 5 UNIT 1 PRACTICE p. 69, #10 6. () c. d. Check students work. 1. ( 2, 5) 2. ( 5, 6) 3. (3, 4) 4. ( 3, -4, 2) 5. Answers will var. consistent and dependent inconsistent CHECK YOUR UNDERSTANDING Write our answers on on notebook paper paper. or grid Show our work. 4. Solve the sstem, using elimination. paper. Show our work z = Solve the sstem b graphing z = -47 { = z = -3 = MATHEMATICAL Which solution method for REFLECTION solving sstems of 2. Solve the sstem, using substitution. equations do ou find easiest to use? Which = method do ou find most difficult to use? { 3 - = -13 Eplain wh. 3. Solve the sstem, using elimination = 17 { 4-2 = 4 20 SpringBoard Mathematics with Meaning Algebra 2 20 SpringBoard Mathematics with Meaning Algebra 2