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1 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Reasoning with Equations and Quantities Name: Date: Understand solving equations as a process of reasoning and eplain the reasoning MCC9-1.A.REI.1 Eplain each step in solving a simple equation as following from the equalit of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justif a solution method. (Students should focus on and master linear equations and be able to etend and appl their reasoning to other tpes of equations in future courses.) Solve equations and inequalities in one variable MCC9-1.A.REI. Solve linear equations and inequalities in one variable, including equations with coefficients represented b letters.(etend earlier work with solving linear equations to solving linear inequalities in one variable and to solving literal equations that are linear in the variable being solved for. Include simple eponential equations that rel onl on application of the laws of eponents, such as 5 = 15 or = 1/16.) Solve sstems of equations MCC9-1.A.REI.5 Prove that, given a sstem of two equations in two variables, replacing one equation b the sum of that equation and a multiple of the other produces a sstem with the same solutions. (Limit to linear sstems.) MCC9-1.A.REI.6 Solve sstems of linear equations eactl and approimatel (e.g., with graphs), focusing on pairs of linear equations in two variables. Represent and solve equations and inequalities graphicall MCC9-1.A.REI.1 Graph the solutions to a linear inequalit in two variables as a half-plane (ecluding the boundar in the case of a strict inequalit), and graph the solution set to a sstem of linear inequalities in two variables as the intersection of the corresponding half-planes. LESSON.0 ALGEBRA EXERCISES Solve for a

2 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page of (n 7) (1 ) 5( 5) (m 7) 7(6 m ) Solve each b factoring 17. ( 1)( ) ( 4)( 5)

3 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page of Lesson.1 Add and Subtract Rational Epressions Eample 1 Find the sum or the difference. a. b. c. d ( 11) ( ) ( ) ( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) PROBLEMS:

4 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 4 of Lesson. Solve Rational Equations Eample 1 Solve Solution: Write the equation 6 6 ( 5) Cross-multipl 6 5 Multipl Bring all the terms on one side 0 ( 9)( 4) Factor 9 or 4 Solve for PROBLEMS:

5 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 5 of Lesson. Goal Vocabular Solve a Linear Sstem b Graphing Solve sstems of linear equations A sstem of linear equations with two variables and consists of two equations that can be written in the following form: A B C and D E F. A solution of a sstem of linear equations with two variables is an ordered pair (, ) that satisfies each equation (plugging in the corresponding numbers for and ). A sstem which has at least one solution is called consistent. The solution point is where both lines cross which was defined earlier as an ordered pair. If there is no solution to a sstem, it is called inconsistent (if both lines are parallel the never touch; therefore there is no solution to the sstem). A consistent sstem with eactl one solution is called independent. A consistent sstem with infinitel man solutions is called dependent: Both lines la on top of each other and ever point is a solution. Eample 1: Solve sstems graphicall 5 Equation1 5 Equation 5 5 Solution: Graph both equations. The lines intersect each other at (, 1), i.e. ordered pair. Check the solution b plugging in the ordered pair into the equations: Equation1 Equation 5 5 ( 1) () 5 ( 1) ()

6 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 6 of PROBLEMS Graph the linear sstem and find its solution. Then check the solution algebraicall

7 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 7 of

8 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 8 of Eample : Use a linear sstem to solve a realistic problem. Car rental compan A charges \$10 per da plus a one-time \$10 rental fee. Car rental compan B charges \$ per da plus a one-time \$50 rental fee. After how man das will the total cost of both options be the same? Solution: Let represent the number of das ou rent the car. Let be the total rental cost Equation1( Car rental compan A) 50 Equation ( Car rental compan B) Cost (\$10/Unit) To solve the sstem graph the equations. The lines intersect at (5,60). Thus after 5 das the total cost (\$60) is the DAYS same at both rental car companies. PROBLEMS 8. The price of car A is \$15000, and the price of car B is \$0000. Car A has earl operating costs of \$000, while car B has earl operating costs of \$000. a. Write an equation for the cost of owning car A and an equation for the cost of owning car B. b. Graph both equations from part a. After how man ears are the total costs of owning the cars equal? Cost (\$) Years c. State a reason wh ou would prefer to own one car over the other.

9 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 9 of 9. John decides to open a bank account. He deposits \$5 and plans to deposit ever week another \$10. Markus has saved \$60 and plans to save \$5 ever week. a. Plot a line which represent John s mone, and plot a line that represents Markus mone over time. Mone Saved (\$) b. When do both lines meet? What does that point mean? Weeks c. Who will have more mone after weeks? d. Who will have more mone after 8 weeks? e. Wh are the slopes of both lines different from each other? f. What does that the slope of each line represent? g. What do the -intercepts of each line represent?

10 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 10 of Lesson.4 Solve Linear Sstem Algebraicall Vocabular Substitution Method: See eample 1 Elimination Method: See eample. Eample 1 Solve the sstem using the substitution method. 6 1 Equation1 5 Equation Solution Step 1: Solve equation for : 5 Equation Step : Substitute the epression for into Equation 1 and solve for : 6 ( 5) 1 Equation Step : Substitute the value of into either equation (we pick Equation ) and solve for. (1) 5 Equation The solution is (1,). PROBLEMS Solve the sstem using the substitution method

11 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 11 of Eample Solve the sstem using the elimination method. 6 1 Equation1 5 Equation Solution Step 1: Multipl equation b -, so that the coefficients of onl differ in sign. Then add both equations and solve for (since is cancelled out b -): 6 1 Equation ( ) Equation 9 15 Step : Substitute the epression for into Equation 1 and solve for : 6 1 Equation1 6(1) 1 6 The solution is (1,). Hint: Alternativel ou could have multiplied Equation b - and cancelled upon addition of both equations. Choose an wa which is most convenient. 1 PROBLEMS Solve the sstem using the elimination method Solve the sstem using an method of our liking

12 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Lesson.5 Graph Linear Inequalities in Two Variables Eample 1 Tell whether the ordered pairs (4, 1) and (,6) are solutions of the inequalit 9. Solution: Substitute both points into the inequalit and check whether the satisf the inequalit. (4) ( 1) 9 ( ) (6) Not True True Therefore, (4, 1) is not a solution but (,6) is a solution. PROBLEMS Tell whether the given ordered pairs are solutions of the inequalit (, 4) (1, ). (0,1) (,1). (0, 1) (,) 4. 4 (1, ) (, 4)

13 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of Eample Graph the linear equation 6. Solution: Solve the equation for. Then graph the line. Plug in a point, sa (0,1). Since the point satisfies the inequalit, the region containing the point is the solution and is shaded. PROBLEMS: Graph the linear equation

14 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 14 of Eample In basketball ou can score b throwing two-point and three-point shots. a. Write an inequalit to represent the number of two-point shots and the number of three-point shots that total at least 15 points. b. Graph the inequalit. Solution a. Let the number of two-point score be, and the number of three-point scores be. Then the number of two-point scores times and the number of three point scores times three have to be equal or eceed 15. The equation for this inequalit can be written as follows: 15 b. In order to plot the equation ou can solve the equation for which ields: 5 The solution is the area on and above the line. (,6) That means an combination of integers for and with a coordinate point on or above the line will be a solution. For eample the point (, 6) is a solution since it is above the line. The score associated with the point is plus 6 which is larger or equal to 15. PROBLEMS: 1. The Boston Celtic coach believes the can win the NBA title if the score at least 90 points per game. a. Write an inequalit to represent the number of two-point shots and the number of three-point shots that total at least 90 points per game. b. Graph the inequalit.

15 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 15 of. In soccer ou receive points for winning and 1 point for a tie. In order to win the national championship a soccer coach estimates his team needs to make at least 50 points. a. Write an inequalit to represent the number of wins and ties that total at least 50 points for the season. b. Graph the inequalit.. You are the owner of a coffee shop and sell espressos for \$ and lattes for \$4. a. Write an inequalit to represent the number of espressos and lattes ou have to sell each da to make a dail revenue of more than \$100. b. Graph the inequalit.

16 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 16 of Lesson.6 Graph Sstems of Linear Inequalities Vocabular An eample of a sstem of linear inequalities is: 6and 4 1. A solution of a sstem of inequalities is an ordered pair that satisfies both inequalities. Eample 1 Graph a sstem of two inequalities. Inequalit1 Inequalit Solution Step 1: Graph each inequalit. Step: Identif the region that is common to both graphs. The region with horizontal and vertical lines is the solution. PROBLEMS: Graph a sstem of two inequalities

17 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 17 of Eample Graph a sstem of three or more inequalities. Inequalit1 Inequalit Inequalit Solution Step 1: Graph each inequalit. Step: Identif the region that is common to all graphs. The region with horizontal and vertical lines is the solution. PROBLEMS: Graph a sstem of inequalities

18 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 18 of. A car compan produces different cars at a plant: A hatchback and a convertible. The maimum production capacit is 0 automobiles per da. Each hatchback generates a profit of \$1000 and each convertible generates a profit of \$000. The compan wants to make at least \$10,000 per da. a. Write an inequalit to represent the maimum production capacit for the plant. b. Write an inequalit to represent the least amount of profit desired per da. c. Write a sstem of inequalities to represent both inequalities above (basicall cop both equations). d. Graph the sstem of inequalities and shade in the solutions. # of Convertibles # of Hatchbacks e. The solution of the sstem is where both areas overlap. What does the area of overlap represent for this eample?

19 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 19 of 4. A small bus can carr a maimum load of 1600 pounds. A child (including luggage) weighs 100 pounds on average and an adult (including luggage) weighs 00 pounds on average. The child s fare is \$1 and the adults fare is \$5. The bus driver wants to make at least \$60 per trip. a. Write an inequalit to represent the most weight the bus driver can have on the bus. b. Write an inequalit to represent the least amount of mone for a trip. c. Write a sstem of inequalities to represent both inequalities above (basicall cop both equations). d. Graph the sstem of inequalities and shade in the solutions. # OF ADULTS # OF CHILDREN e. The solution of the sstem is where both areas overlap. What does the area of overlap represent for this eample?

20 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 0 of 5. You are shopping for a bike. The regular price of a bike is between \$00 and \$400. The store is running a sale where all bikes are between 0% and 60% off the regular price. The sstem of linear equations represents the amount ou can save. Let represent the regular price. Let represent the amount ou can save The regular price is between \$00 and \$400 The sales price is between 0% (=0.0) and 60% (=0.60) of the regular price a. Plot the inequalit SAVINGS (\$) REGULAR PRICE (\$) b. What is the most ou can save? (Hint: Look at the point (400,40)) c. What is the least ou can save? (Hint: Look at the point (00,40)) d. What is the most ou will pa for the most epensive bike? (Hint: Look at the point (400,80)) e. What is the most that ou will pa for the least epensive bike? (Hint: Look at the point (00,40)) f. What is the least ou will pa for the most epensive bike? (Hint: Look at the point (400,80))

21 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page 1 of 6. A compan is manufacturing computers. A tablet computer costs \$00 to make and a laptop costs \$600 to make. The budget available for materials is \$1,000 per da. The manufacturing capacit is 50 computers per da. Let represent the tablets. Let represent the laptops. a. Write a sstem of linear inequalities to represent the problem situation. b. Graph the inequalit. # OF LAPTOPS # OF TABLETS

22 CCGPS UNIT Semester 1 COORDINATE ALGEBRA Page of Lesson.7 Graph Sstems of Linear Inequalities Using Technolog You can use a graphing calculator (TI-8) to graph a sstem of linear inequalities. STEP 1: Press Y= and enter the two inequalities as Y₁ and Y₂. STEP : While in the Y= mode, access the inequalit function b moving the cursor to the left until the \ flashes. Press ENTER to select the appropriate inequalit: Flashing triangle on right top: Y> Flashing triangle on left bottom: Y< STEP : Step 4: Press Window to set the bounds. Press Graph. PROBLEMS: Graph a sstem of inequalities using a graphing calculator

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