Using Systems to Solve Problems ( day 3 of 3 ) MCC9-12.A.REI.5 & MCC9-12.A.REI6

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=____________________ # of $20 dollar bills # of $10 dollar bills Use the total number of bills to write one equation: ______+______=________ (total tells us to add) x y 12 Use the value to write another equation: ________+________=_________ 20x 10y 170

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

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. l = length 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_____________________________________ 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? Substitution. Solve for ‘x’ x + y = 4000 x = amount invested in 4% CD ( ) .04x + .06y = 188 y = amount invested in 6% CD ( ) $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 advance y = tickets sold at door Elimination ( ) x + y = 100 ( ) -10 -10 -10x - 10y = -1000 10x + 15y = 1200 + 10x + 15y = 1200 5y = 200 y = 40 x + 40 = 100 x = 60 40 tickets at the door 60 tickets in advance

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 walking y = hours spent jogging Elimination 2x + 4y = 3.4 2x + 4y = 3.4 -2x - 2y = -2.4 ( ) x + y = 1.2 ( ) + -2 -2 2y = 1 y = .5 or 1/2 x + .5 = 1.2 .7 hours or 42 minutes walking .5 hours or 30 minutes jogging x = .7