ALGEBRA UNIT 5 -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 Find the solutions to the following system of equations graphically then determine whether each system is consistent or inconsistent and if it is independent or dependent: 1. y 2x 8 y x 2 1
2. x 2y 8 2x 4y 4 3. 2 1 1 x y 3 6 2 3 1 1 x y 4 2 4 TO CHECK ON YOUR GRAPHING CALCULATOR (FIND INTERSECTION): 2nd Trace # 1) Go to (Calculate) and pick (intersection) 2) Move cursor to wanted intersection point Enter Enter Enter 4. Use your calculator to check the previous examples. 5. Use your calculator to solve the following systems of equations: 5a. 2x - 2y = 5 5b. x 2y = -4 4x 4y = 9 4y = 2x +8 2
SOLVING SYSTEMS ALGEBRAICALLY (DAY 2) METHOD 1: ELIMINATIION METHOD 2: SUBSTITUTION 1. Rearrange into standard form Ax + By = C 1. Make sure at least one of the equations is in x = form or y = form 2. Look at coefficients. You want one of the variables to be the same value but the opposite sign (sometimes you have to multiply one or both equations by a number to get this to this). 3. By combining (adding) the 2 equations together you will eliminate one variable. 4. Solve for the remaining variable. 5. Once you find one variable go back and find the other by substituting the value into one of the original equations. 2. Box one equation that is represented by a variable. Substitute this boxed part into the other equation for the variable it represents, to get all the same variables in the new equation. 3. Solve for the missing variable. 4. Once you find one variable go back and find the other by substituting the value into one of the original equations. 5. Check both solutions in EACH equation. 6. Check both solutions in EACH equation. Solve the following systems algebraically: 1. 2x 3y 26 5y 2x 22 2. f 2g 1 5g 2f 2 3
3. 11x 7y 14 x 2y 4 4. 8m 3r 10 10m 2r 2 4
SOLVING SYSTEMS ALGEBRAICALLY cont.. (DAY 3) Solve the following systems algebraically: 1. 2 1 1 x y 2. 3 6 2 3 4 1 x y 2 1 4 x 3 5 2 3 y 4 x y 15 5
SOLVING SYSTEMS ALGEBRAICALLY WORD PROBLEMS (DAY 4) 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? 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? 6
4. Lilly and Rosie are sisters. The sum of their ages is 19 and the positive difference of their ages is 9. Set up a system of equations involving Lilly s age, L, and Rosie s age, R, assuming that Lilly is the older child. Solve the system to find their ages. 5. The measure of the larger of two complementary angles is 6 less than twice the measure of the smaller angle. Find the degree measure of each angle. Complementary: Angles add up to Supplementary: Angles add up to 7
SOLVING SYSTEMS GRAPHICALLY WORD PROBLEMS (DAY 5) 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. In function notation, write A(x) to represent the total cost of attending carnival A and going on x rides. In function notation, write B(x) to represent the total cost of attending carnival B and going on x rides. b. Determine the number of rides Maggie can go on such that the total cost of attending each carnival is the same. [Use the set of axes below] c. Maggie wants to go on five rides. Determine which carnival would have the lower total cost. Justify your answer. 8
2. 3. You are purchasing jeans and T-shirts. Jeans cost $35 and T-shirts cost $15. You only have $115 to spend and plan on purchasing a total of 5 items. a. Write a system of equations that models the situation. b. Solve the system graphically. Show appropriate scale and label axes appropriately. c. How many pairs of jeans and how many T-shirts can you buy? d. Explain why a point in the fourth quadrant does not satisfy the system. 9
SOLVING SYSTEMS: MONEY/PERCENT WORD PROBLEMS (Day 6) 1. Your cousin borrowed $6000, some on home-equity loan at an interest rate of 11% and the rest on a computer loan at an interest rate of 9.5%. Her total interest paid was $645. How much did she borrow at each rate? 2. 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? 4. A purse contains $7.60 in quarters and dimes. In all, there are 40 coins. How many of each kind are there? 10
SOLVING SYSTEMS OF 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 11
3. 3 y x 4 5 3x 5y 5 4. Graph the system of linear inequalities: a.) Describe the shape of the solution region b.) Find the vertices of the solution region c.) Find the area of the solution region 12
SOLVING SYSTEMS OF INEQUALITIES Word Problems (Day 8) 1. For a hiking trip, you are making a mix of x ounces of peanuts and y ounces of chocolate pieces. You want the mix to have less than 70 grams of fat and weigh less than 8 ounces. An ounce of peanuts has 14 grams of fat, and an ounce of chocolate pieces has 7 grams of fat. Write and graph a system of inequalities that models the situation. 2. You are fishing in a marina for perch and rockfish, which are two species of bottomfish. Gaming laws in the marina allow you to catch no more than 15 perch, no more than 10 rockfish per day, and no more than 15 total bottomfish per day. Write and graph a system of inequalities that model the situation. 13
3. Mike makes $7 an hour working at the grocery store and $10 an hour delivering newspapers. He cannot work more than 20 hours per week. Graph two inequalities that Mike can use to determine how many hours he needs to work at each job if he wants to earn at least $90. 14