1. Using Systems to Solve Problems (day 3 of 3) MCC9-12.A.REI.5 & MCC9-12.A.REI6 Learning Target: I am learning to write and solve a system of equations to answer real world problems. In Unit 1, you wrote and solved equations with one variable. In this lesson, we will write and solve a system of 2 equations using 2 different variables to answer a real world problem.

2. To solve a word problem by using a system of 2 equations with 2 variables: 1. Decide what 2 quantities you need to find and choose 2 variables to represent them. (This is referred to as “defining your variables”.) 2. Use the information given in the problem to write two equations which show the relationships between the quantities. 3. Solve the system ( use substitution, elimination, or graphing). 4. Interpret the solution to answer the problem. 5. Check your answer in the original question.

3. Ex. 1 When Angelo cashed a check for $170, the bank teller gave him 12 bills, some $20s and the rest $10s. How many of each denomination did Angelo receive? We need to determine how many $20 bills and how many $10 bills he received Define variables: x=____________________ y=____________________ Use the total number of bills to write one equation: ______+______=________ (total tells us to add) Use the value to write another equation: ________+________=_________ # of $20 dollar bills # of $10 dollar bills x y 12 20x 10y 170

4. Solve the system: Interpret your solution, and check in the original problem. Angelo received 5 twenties and 7 tens. Substitution. Solve for ‘x’ ( )

5. Ex. 2: A rectangle has a perimeter of 38ft. The length is 1ft less than 3 times the width. Find the dimensions of the rectangle. Define variables: ____________ _____________ Write an equation that relates the length and the width. _______________________ Recall the formula for the perimeter of a rectangle(or just add all the sides) to write an equation. _______________________________________ or_____________________________________ l = length w = width l = 3w - 1 p = l + l + w + w p = 2l + 2w

6. Solve the system. Interpret your solution to answer the question, and check your answer. Substitution. ‘l’ is already solved for!! ( ) A rectangle with length 14 and width 5.

7. Ex. 3. Mr. Stamos invested a sum of money in CDs earning 4% a year and another sum of money in bonds earning 6% a year. The total he invested was $4000. If his combined interest earned for the year was $188, how much did he invest in each? x = amount invested in 4% CD y = amount invested in 6% CD x + y = 4000.04x +.06y = 188 Substitution. Solve for ‘x’ ( ) $2600.00 at 4% $1400.00 at 6%

8. Ex. 4 Tickets for a dance cost $10 in advance and $15 at the door. If 100 tickets were sold and $1200 was collected, how many tickets were sold in advance and how many were sold at the door? x = tickets sold in advancey = tickets sold at door x + y = 100 10x + 15y = 1200 Elimination ( ) 40 tickets at the door 60 tickets in advance ( ) -10 -10x - 10y = -1000 10x + 15y = 1200 + 5y = 200 y = 40 x + 40 = 100 x = 60

9. Ex. 5. Carmen walks at a rate of 2 mph and jogs at a rate of 4mph. she walked and jogged a total of 3.4 miles in 1.2 hours. For how long did Carmen jog, and for how long did she walk? x = hours spend walkingy = hours spent jogging 2x + 4y = 3.4 x + y = 1.2 Elimination ( ) -2 2x + 4y = 3.4 -2x - 2y = -2.4 + 2y = 1 y =.5 or 1/2 x +.5 = 1.2 x =.7.7 hours or 42 minutes walking.5 hours or 30 minutes jogging