Sstems of Linear Equations Monetar Sstems Overload Lesson 3-1 Learning Targets: Use graphing, substitution, and elimination to solve sstems of linear equations in two variables. Formulate sstems of linear equations in two variables to model real-world situations. SUGGESTED LEARNING STRATEGIES: Shared Reading, Close Reading, Create Representations, Discussion Groups, Role Pla, Think- Pair-Share, Quickwrite, Note Taking, Look for a Pattern Have ou ever noticed that when an item is popular and man people want to bu it, the price goes up, but 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 + 9.7 Suppl function: P = 1.Q 10. 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. 1 10 p CONNECT TO ECONOMICS The role of the desire for and availabilit of a good in determining price was described b Muslim scholars as earl as the fourteenth centur. The phrase suppl and demand was first used b eighteenth-centur Scottish economists. 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. Price (dollars) 8 10 Q Gasoline (millions of barrels) 1. Make use of structure. Find an approimation of the coordinates of the intersection of the suppl and demand functions. Eplain what the point represents. Sample answer: (9.1, 3.3); At a price of $3.30, people will demand 9.1 million gallons of gas, and companies will be willing to suppl it.
Lesson 3-1 TECHNOLOGY TIP You can use a graphing calculator and its Calculate function to solve sstems of equations in two variables.. What problem(s) can arise when solving a sstem of equations b graphing? Sample answer: Graphing is not ver accurate if the intersection is not on a lattice point, or the scaling of the graph is not accurate enough. 3. Model with mathematics. For parts a c, graph each sstem. Determine the number of solutions. = + a. 1 = + one solution = + b. = no solutions Sstems of linear equations are classified b the number of solutions. Sstems with one or man solutions are consistent. Sstems with no solution are inconsistent. A sstem with eactl one solution is independent. A sstem with infinite solutions is dependent. = + c. 1 = + infinitel man solutions d. Graphing two linear equations illustrates the relationships of the lines. Classif the sstems in parts a c as consistent and independent, consistent and dependent, or inconsistent. a. consistent and independent b. inconsistent c. consistent and dependent
Lesson 3-1 Check Your Understanding. Describe how ou can tell whether a sstem of two equations is independent and consistent b looking at its graph.. The graph of a sstem of two equations is a pair of parallel lines. Classif this sstem. Eplain our reasoning.. Make sense of problems. A sstem of two linear equations is dependent and consistent. Describe the graph of the sstem and eplain its meaning. 7. Marlon is buing a used car. The dealership offers him two pament plans, as shown in the table. Plan Pament Plans Down Pament ($) Monthl Pament ($) 1 0 300 300 00 Marlon wants to answer this question: How man months will it take for him to have paid the same amount using either plan? Work with our group on parts a through f and determine the answer to Marlon s question. a. Write an equation that models the amount Marlon will pa to the dealership after months if he chooses Plan 1. = 300 b. Write an equation that models the amount Marlon will pa to the dealership after months if he chooses Plan. = 300 + 00 CONNECT TO PERSONAL FINANCE A down pament is an initial pament that a customer makes when buing an epensive item, such as a house or car. The rest of the cost is usuall paid in monthl installments. DISCUSSION GROUP TIP As ou work with our group, review the problem scenario carefull and eplore together the information provided and how to use it to create a potential solution. Discuss our understanding of the problem and ask peers or our teacher to clarif an areas that are not clear. c. Write the equations as a sstem of equations. = 300 { = 300 + 00
Lesson 3-1 d. Graph the sstem of equations on the coordinate grid. MATH TIP When graphing a sstem of linear equations that represents a real-world situation, it is a good practice to label each line with what it represents. In this case, ou can label the lines Plan 1 and Plan. Total Amount Paid ($) 100 100 10800 9000 700 00 300 1800 Used Car Pament Plans Plan Plan 1 1 18 30 3 Time (months) e. Reason quantitativel. What is the solution of the sstem of equations? What does the solution represent in this situation? (3, 10,800); In 3 months, the total cost of both plans will be $10,800. f. In how man months will the total costs of the two plans be equal? 3 months Check Your Understanding 8. How could ou check that ou solved the sstem of equations in Item 7 correctl? 9. If Marlon plans to keep the used car less than 3 ears, which of the pament plans should he choose? Justif our answer. 10. Construct viable arguments. Eplain how to write a sstem of two equations that models a real-world situation.
Lesson 3-1 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. Eample A During one ear, Sara invested $000 into two separate funds, one earning percent and another earning percent annual interest. The interest Sara earned was $0. How much mone did she invest in each fund? Step 1: 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. + = 000 The mone invested is $000. 0.0 + 0.0 = 0 The interest earned is $0. Step : Use substitution to solve this sstem. + = 000 Solve the first equation for. = 000 0.0 + 0.0(000 ) = 0 Substitute for in the second equation. 0.0 + 0 0.0 = 0 Solve for. 0.03 = = 100 Step 3: Substitute the value of into one of the original equations to find. + = 000 100 + = 000 Substitute 1,00 for. = 300 Solution: Sara invested $100 in the first fund and $300 in the second fund. Tr These A Write our answers on notebook paper. Show our work. Solve each sstem of equations, using substitution. Check students work. = a. 3 + = b. + = 9 1 = c. = 10 3 + = 1 ( 1, 13) (1, 1) (3, 7) d. Model with mathematics. Eli invested a total of $000 in two stocks. One stock cost $18.0 per share, and the other cost $10.0 per share. Eli bought a total of 130 shares. Write and solve a sstem of equations to find how man shares of each stock Eli bought. + = 130 { 18. + 10. = 000 (80, 0); 80 shares at $18.0, 0 shares at $10.0 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 value of the other variable. MATH TIP Check our answer b substituting the solution (100, 300) into the second original equation, 0.0 + 0.0 = 0
Lesson 3-1 11. When using substitution, how do ou decide which variable to isolate and which equation to solve? Eplain. 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. Another algebraic method for solving sstems of linear equations is the elimination method. 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. The elimination method is also called the addition-elimination method or the linear combination method for solving a sstem of linear equations. Eample B A stack of 0 coins contains onl nickels and quarters and has a total value of $. 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. n + q = 0 The number of coins is 0. n + q = 00 The total value is 00 cents. Step : To solve this sstem of equations, first eliminate the n variable. (n + q) = (0) Multipl the first equation b. n + q = 00 n q = 100 n + q = 00 Add the two equations to eliminate n. 0q = 300 Solve for q. q = 1 Step 3: Find the value of the eliminated variable n b using the original first equation. n + q = 0 n + 1 = 0 Substitute 1 for q. n = Step : Check our answers b substituting into the original second equation. n + q = 00 () + (1) = 00 Substitute for n and 1 for q. + 37 = 00 00 = 00 Solution: There are nickels and 1 quarters in the stack of coins.
Lesson 3-1 Tr These B Solve each sstem of equations using elimination. Show our work. Check students work. = a. 3 + = 1 b. + 3 = 0 + = c. = 10 3 3 1 = 17 (, ) (, ) ( 3, ) d. A karate school offers a package of 1 group lessons and private lessons for $110. It also offers a package of 10 group lessons and 3 private lessons for $1. Write and solve a sstem of equations to find the cost of a single group lesson and a single private lesson. 1g + p = 110 ; (, ); $ for a group lesson, $ for a private { 10g + 3p = 1 lesson Check Your Understanding 1. Compare and contrast solving sstems of equations b using substitution and b using elimination. = 13. Reason abstractl. T is solving the sstem 8 + = 10 using substitution. He will start b solving one of the equations for. Which equation should he choose? Eplain our reasoning. 1. Eplain how ou would eliminate one of the variables in this sstem: = 1. 3 + = 9 LESSON 3-1 PRACTICE 1. Solve the sstem b graphing. 1. Solve the sstem using substitution. + 9 = = 3 + 19 = 3 = 13 3 + = 17 17. Solve the sstem using elimination. = 18. Make sense of problems and persevere in solving them. At one compan, a level I engineer receives a salar of $,000, and a level II engineer receives a salar of $8,000. The compan has 8 level I engineers. Net ear, it can afford to pa $7,000 for their salaries. Write and solve a sstem of equations to find how man of the engineers the compan can afford to promote to level II. 19. Which method did ou use to solve the sstem of equations in Item 18? Eplain wh ou chose this method.