Systems word problems notes
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This document provides examples for solving word problems by writing equations and using substitution or elimination methods. It includes 12 examples of common word problems that involve defining variables, writing equations, selecting a solution method, and providing the answer. The examples cover a variety of problem types involving rates, mixtures, age problems, geometry, financial problems, and more.
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Systems word problems notes
- 1. Solving Systems Using Word Problems
- 2. Objectives • Use reading strategies to write formulas • Solve the equations using substitution or elimination
- 3. Steps to Follow • 1. Identify Variables • 2. Write 2 equations • (Use key words and reading strategies to help) • 3. Solve using substitution or elimination • 4. Write answer in a complete sentence
- 4. Example 1Example 1 Kevin would like to buy 10 bouquets. The standardKevin would like to buy 10 bouquets. The standard bouquet costs $7, and the deluxe bouquet costs $12. Hebouquet costs $7, and the deluxe bouquet costs $12. He can only afford to spend $100. How many of each typecan only afford to spend $100. How many of each type can he buy?can he buy? Define Variables:Define Variables: X: standard bouquet Y: deluxe bouquetX: standard bouquet Y: deluxe bouquet Equation 1 Cost:Equation 1 Cost: 7x+12y=1007x+12y=100 Equation 2 Bouquets :Equation 2 Bouquets : x+y=10x+y=10 Best Method :Best Method : EliminationElimination Solution: (4,6)Solution: (4,6)
- 5. Example 2Example 2 A hot air balloon is 10 meters above the ground and risingA hot air balloon is 10 meters above the ground and rising at a rate of 15 meters per minute. Another balloon is 150at a rate of 15 meters per minute. Another balloon is 150 meters above the ground and descending at a rate of 20meters above the ground and descending at a rate of 20 meters per minute. When will the two balloons meet?meters per minute. When will the two balloons meet? Define Variables:Define Variables: x=minutes y=height in metersx=minutes y=height in meters Equation 1Equation 1 y=15x+10y=15x+10 Equation 2Equation 2 y=-20x+150y=-20x+150 Best Method:Best Method: SubstitutionSubstitution Solution: (4,70)Solution: (4,70)
- 6. Example 3Example 3 A group of 3 adults and 10 students paid $102 for aA group of 3 adults and 10 students paid $102 for a cavern tour. Another group of 3 adults and 7 studentscavern tour. Another group of 3 adults and 7 students paid $84 for the tour. Find the admission price for anpaid $84 for the tour. Find the admission price for an adult ticket and a student ticket.adult ticket and a student ticket. Define Variables:Define Variables: x= adult ticket price y=student ticket pricex= adult ticket price y=student ticket price Equation 1Equation 1 3x+10y=1023x+10y=102 Equation 2Equation 2 3x+7y=843x+7y=84 Best MethodBest Method EliminationElimination Solution (14,6)Solution (14,6)
- 7. Example 4Example 4 Melissa and Frank were jogging. Melissa had a 2Melissa and Frank were jogging. Melissa had a 2 mile head start on Frank. If Melissa ran at an average ratemile head start on Frank. If Melissa ran at an average rate of 5 miles per hour and Frank ran at an average rate of 8of 5 miles per hour and Frank ran at an average rate of 8 miles per hour, how long would it take for Frank to catchmiles per hour, how long would it take for Frank to catch up with Melissa?up with Melissa? Define Variables:Define Variables: x=hours y=milesx=hours y=miles Equation 1Equation 1 y=5x+2y=5x+2 Equation 2Equation 2 y=8xy=8x Best MethodBest Method SubstitutionSubstitution Solution (2/3, 5 1/3) orSolution (2/3, 5 1/3) or (2/3, 16/3)(2/3, 16/3)
- 8. Example 5Example 5 An Algebra Test contains 38 problems. Some of theAn Algebra Test contains 38 problems. Some of the problems are worth 2 points each. The rest of theproblems are worth 2 points each. The rest of the questions are worth 3 points each. A perfect score isquestions are worth 3 points each. A perfect score is 100 points. How many problems are worth 2 points?100 points. How many problems are worth 2 points? How many problems are worth 3 points?How many problems are worth 3 points? Define Variables:Define Variables: x=2 pt. questions y=3 pt. questionsx=2 pt. questions y=3 pt. questions Equation 1Equation 1 x+y=38x+y=38 Equation 2Equation 2 2x+3y=1002x+3y=100 Best MethodBest Method Elimination or SubstitutionElimination or Substitution Solution (14,24)Solution (14,24)
- 9. Example 6Example 6 Ashley has $9.05 in dimes and nickels. If she hasAshley has $9.05 in dimes and nickels. If she has a total of 108 coins, how many of each type doesa total of 108 coins, how many of each type does she have?she have? Define VariablesDefine Variables x=dimes y=nickelsx=dimes y=nickels Equation 1Equation 1 x+y=108x+y=108 Equation 2Equation 2 .10x+.05y=9.05.10x+.05y=9.05 Best MethodBest Method SubstitutionSubstitution Solution (73,35)Solution (73,35)
- 10. Example 7Example 7 The perimeter of a parking lot is 310 meters. TheThe perimeter of a parking lot is 310 meters. The length is 10 more than twice the width. Find thelength is 10 more than twice the width. Find the length and width. (Remember: P=2L+2W)length and width. (Remember: P=2L+2W) Define VariablesDefine Variables L=length W=widthL=length W=width Equation 1Equation 1 2L+2W=3102L+2W=310 Equation 2Equation 2 L=2W+10L=2W+10 Best MethodBest Method SubstitutionSubstitution Solution (106 2/3, 48 1/3)Solution (106 2/3, 48 1/3)
- 11. Example 8Example 8 The sum of two numbers is 112. The smaller is 58 less thanThe sum of two numbers is 112. The smaller is 58 less than the greater. Find the numbers.the greater. Find the numbers. Define VariablesDefine Variables x=smaller number y=larger numberx=smaller number y=larger number Equation 1Equation 1 x+y=112x+y=112 Equation 2Equation 2 x=y-58x=y-58 Best MethodBest Method SubstitutionSubstitution Solution (27,85)Solution (27,85)
- 12. Example 9Example 9 The sum of the ages of Ryan and his father is 66. HisThe sum of the ages of Ryan and his father is 66. His father is 10 years more than 3 times as old as Ryan.father is 10 years more than 3 times as old as Ryan. How old are Ryan and his father?How old are Ryan and his father? Define VariablesDefine Variables x=Ryan’s age y=Dad’s agex=Ryan’s age y=Dad’s age Equation 1Equation 1 x+y=66x+y=66 Equation 2Equation 2 y=3x+10y=3x+10 Best MethodBest Method SubstitutionSubstitution Solution (14,52)Solution (14,52)
- 13. Example 10Example 10 A total of $10,000 is invested in two funds, Fund A andA total of $10,000 is invested in two funds, Fund A and Fund B. Fund A pays 5% annual interest and Fund BFund B. Fund A pays 5% annual interest and Fund B pays 7% annual interest. The combined annual interest ispays 7% annual interest. The combined annual interest is $630. How much of the $10,000 is invested in each fund?$630. How much of the $10,000 is invested in each fund? Define VariablesDefine Variables a=Fund A b=Fund Ba=Fund A b=Fund B Equation 1Equation 1 a+b=10,000a+b=10,000 Equation 2Equation 2 .05a+.07b=630.05a+.07b=630 Best MethodBest Method SubstitutionSubstitution Solution (6500,3500)Solution (6500,3500)
- 14. Example 11Example 11 We need to rent a large truck for one week.We need to rent a large truck for one week. Rental companies charge an initial cost plus anRental companies charge an initial cost plus an additional cost for each mile driven. One company,additional cost for each mile driven. One company, Paenz, will rent a 27 foot truck for us for $525 plusPaenz, will rent a 27 foot truck for us for $525 plus $0.21 per mile. Another company, Opan, will rent us$0.21 per mile. Another company, Opan, will rent us the same size truck for $585 plus $0.13 per mile.the same size truck for $585 plus $0.13 per mile. Define VariablesDefine Variables x=miles y=total costx=miles y=total cost Equation 1Equation 1 y=0.21x+525y=0.21x+525 Equation 2Equation 2 y=0.13x+585y=0.13x+585 Best MethodBest Method SubstitutionSubstitution Solution (750,682.50)Solution (750,682.50)
- 15. Example 12Example 12 The larger of two numbers is 7 less than 8 times theThe larger of two numbers is 7 less than 8 times the smaller. If the larger number is decreased by twicesmaller. If the larger number is decreased by twice the smaller, the result is 329. Find the two numbers.the smaller, the result is 329. Find the two numbers. Define VariablesDefine Variables x=smaller number y=larger numberx=smaller number y=larger number Equation 1Equation 1 y=8x-7y=8x-7 Equation 2Equation 2 y-2x=329y-2x=329 Best MethodBest Method SubstitutionSubstitution Solution (56,441)Solution (56,441)