ALGEBRA UNIT 5 LINEAR SYSTEMS SOLVING SYSTEMS: GRAPHICALLY (Day 1)

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1 ALGEBRA UNIT 5 LINEAR SYSTEMS SOLVING SYSTEMS: GRAPHICALLY (Day 1) System: Solution to Systems: Number Solutions Exactly one Infinite No solution Terminology Consistent and Consistent and Inconsistent independent dependent Graph Procedure for Solving Equations Graphically 1. Graph each equation on the same set of axes 2. Circle the points where the graphs intersect. These are your solutions. 3. Write the solution as an ordered pair in parentheses (x, y). 4. Check your solutions using your calculator 1) Solve the following system graphically y 2x 8 y x 2 TO CHECK ON YOUR GRAPHING CALCULATOR 1) Go to 2nd Trace (Calculate) and pick #5 (intersection) 2) Move cursor to wanted intersection point and hit ENTER ENTER ENTER 1

2 2. You are looking to join a monthly coffee delivery club. Your search has been narrowed down to two top rated clubs on the internet. Java Coffee House charges a $40 startup fee and $6 per month. Two-Cents Coffee Company charges a $25 startup fee and $10.50 per month. a) Write a function for each coffee company. b) Using the grid sketch each function c) During what interval time frame is the Java Coffee house the better option? 3. Solve the following system graphically x y 2 x 1 y

3 SOLVING LINEAR SYSTEMS ALGEBRAICALLY (DAY 2) METHOD 1: Use this Method if... METHOD 2: Use this Method if... x and y variables are on the same side Of the equal sign 1-variable is = to something (i.e. x = or y = ) Equations line up so variables are underneath each other. What you want to happen How to actually solve the system!!! You want a set of numbers in front of a variable to be the but signs. (You may have to multiply one or both equations by a number to get the variables to do this.) How to actually solve the system!!! Add the 2 equations together in order to 1- variable so you can and solve for the remaining variable in the equation After 1- variable is solved for, plug your answer back into 1 of the equations to solve for the 2 nd variable. Check your solutions Box the something part that the variable equals Draw an arrow to the other equation where this boxed part will be plugged into. boxed part into the equation and solve for the remaining variable. After 1- variable is solved for, plug your answer back into 1 of the equations to solve for the 2 nd variable. Check your solutions Solve the following systems algebraically: 1. 2x 3y 26 5y 2x f 2g 1 5g 2f 2 3

4 3. 1 3x y 3 8 2x 3y x 5y 18 3x y a b 5 a 2b 7 4

5 SOLVING LINEAR SYSTEMS ALGEBRAICALLY GROUP WORK-DAY 3 Solve the following systems algebraically. 1. y 4x 1 1 y x x y 5 x 4y y x x 4.5y x y y x

6 5. What is the solution of the system of equations below? 2x 3y 7 x y 3 (1) (1, 2) (2) (2, 1) (3) (4, -1) (4) (4, 1) 6. What is the value of x in the solution of the system of equations 3x 2y 12 and 5x 2y 4? (1) 8 (2) 2 (3) 3 (4) 4 Solve the following systems algebraically 2x 2y y 2x 3 2 x y x y 8 3 6

7 LINEAR SYSTEM APPLICATION WORD PROBLEMS (DAY 4) How to solve/set up word problem systems: Read problem carefully determine the totals o 1 total NOT A SYSTEM (only 1 variable will be in the equation) o 2 totals SYSTEM ( At least 2 variable will be in the equations) Determine what the totals represent to help you create the equations o Ex. Total of 30 coins (equation should just add the coins) o Ex. Total of $4.80 from coins (equations must have $$ value by variables) After equations are set up, determine the correct method to solve either elimination or substitution method) 1. At a quick lunch counter, 6 pretzels and 2 cup of soda costs $5.50. Two pretzels and 1 cup of soda cost $2.00. Find the cost of a pretzel and the cost of a cup of soda. 2. Your family is planning a 7-day trip to Florida. You estimate that it will cost $275 per day in Tampa and $400 per day in Orlando. Your total budget for the 7 days is $2300. How many days should you spend in each location? 7

8 3. A purse contains $7.60 in quarters and dimes. In all, there are 40 coins. How many of each kind are there? 4. Your cousin borrowed $6000, some on a home-equity loan at an interest rate of 11% and the rest on a computer loan at and interest rate of 9.5%. Her total interest paid was $645. How much did she borrow at each rate? 5. Irene has $4.50 in her coin bank in nickels and dimes. The number of dimes is twice the number of nickels. How many coins of each type does Irene have? 8

9 Lin LINEAR APPLICATION WP GROUP WORK (Day 5) 1. During its first week of business, a market sold a total of 108 apples and oranges. The second week, five times the number of apples and three times the number of oranges were sold. A total of 452 apples and oranges were sold during the second week. Determine how many apples and how many oranges were sold the first week. 2. Joe and Caleb together have 28 movies to sell for the yard sale. If Joe has four more movies than Caleb, how many movies does each person have to sell? 3. At a bakery, one customer pays $5.67 for 3 bagels and 4 muffins. Another customer pays $6.70 for 5 bagels and 3 muffins. How much does a single bagel and a single muffin cost? 9

10 4. Cody invested a sum of money in a certificate of deposit yielding 5% a year and another sum in bonds yielding 7% a year. A total of $10,000 is invested. If the combined annual income is $644, how much of the $10,000 did Cody invest at each rate? 5. Marissa has $2.50. She has seven more dimes than nickels. How many of each coin does she have? 6. A collection of 33 coins, consisting of nickels, dimes, and quarters, has a value of $3.30. There are three times as many nickels as quarters, and one-half as many dimes as nickels. How many of each coin are there? 10

11 APPLICATION SYSTEM WP SOLVED GRAPHICALLY (DAY 6) 1. Next weekend Maggie wants to attend either carnival A or carnival B. Carnival A charges $6 for admission and an additional $1.50 per ride. Carnival B charges $2.50 for admission and an additional $2 per ride. a. Write two function A(x) and B(x) to model the situation described above, where x represents the number of rides. b. Using the grid, sketch each function. c. Which Carnival should Maggie choose? Justify your reasoning. d. Determine the number of rides Maggie can go on such that the total cost of attending each carnival is the same. 11

12 2. Tammy needs to go school clothes shopping. She wants to buys some more jeans and t-shirts. At the store the jeans cost $35 and T-shirts cost $15. She only has $115 to spend and is planning on purchasing a total of 5 items. a. Write a system of equations that models the situation. b. Using the grid, sketch each equation. c. Tammy found 4 pairs of jeans and 1 top she really likes. Can she purchase them? Explain your answer. 12

13 SOLVING SYSTEMS OF LINEAR INEQUALITIES (Day 7) There are TWO additional things to think about when graphing inequalities. They are and. RECALL: to determine which side of the line to shade, pick a To identify the solution, place an S in the area that is double shaded. To check your answers, choose a point that falls within the and then substitute it back into both of the equations (it must come out TRUE for both) Graph the following system of inequalities, and indicate the solution. Check your answers. If there is no solution, then write no solution. ` x 2 1. y 2 2. x y 4 y 2x 3 13

14 3. 3 y x 4 5 3x 5y 5 4. Graph the system of linear inequalities: x 2, y 3, andx y 4on the grid below a) Describe the shape of the solution region b) Find the vertices of the solution region c) Find the area of the solution region 14

15 LINEAR INEQUALITY APPLICATION WP (Day 8) 1. Stacie is on the homecoming dance committee and has been but in charge of snacks. She wants to buy an assortment of cheese and crackers. One package of gourmet crackers cost $3.00. The cheese cost $5.00 per pound. She has been allotted $50.00 total for the snacks, so she would like to spend less to conserve on money. After talking to the advisors Stacie knows that she will buy at least 4 pounds of cheese to serve at the dance. a. Create two equations to represent the situation. b. Using the grid, sketch each equation and identify 2 possible combinations of boxes of crackers and pounds of cheese that Stacie can purchase. c. Is the point (-3, 7) a possible solution? Explain your reasoning. 15

16 2) A clothing manufacturer has 1000 yards of cotton to make shirts and pajamas. A shirt requires 1 yard of fabric and pair of pajamas requires 2 yards of fabric. It takes 2 hours to make a shirt and 3 hours to make pajamas, and there are 1600 hours available to make the clothing. a) Write and graph a system of inequalities that models the situation. b) The manufacturer will be selling the shirts for $10 and the pajamas for $14. For a maximum profit what should the manufacturer sell? Explain your reasoning? c) A store orders both the shirts and pajamas to be sold to be sold for the same price. How could they maximize their profit if they must sell both items? 16