Lesson OVERVIEW of lessons on solving systems of two linear equations in two unknowns

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OVERVIEW of lessons on solving systems of two linear equations in two unknowns

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Word problems that lead to a simple system of two equations in two unknowns

      Problem 1. Colin is 4 years older than Emily. The sum of their ages is 34.
                          How old each person is ?

      Problem 2. There are dimes and quarters in a jar. The total number of coins in the jar is 25.
                          The number of dimes is 7 more than the number of quarters. How many coins of each nomination are there in the jar ?

      Problem 3. A rope of the length 20 meters is cut in two pieces.
                          One piece is 2 meters longer than the other. Find the length of each piece.

      Problem 4. One box is 5 pounds havier than the other box.
                          The total weight of both boxes is 17 pounds. Find the weight of each box.

      Problem 5. The adult ticket costs 7 dollars more than the child ticket.
                          Both tickets together cost 30 dolllars. Find the price of each ticket.

      Problem 6. A baseball team played 172 complete games last season.
                          They had 20 fewer wins than losses. How many games did the team win?

      Problem 7. According to statistics, a person will devote 33 years to sleeping and watching TV.
                          The number of years sleeping will exceed the number of years watching TV by 23.
                          Over the lifetime, how many years will the person spend on each of these activities?

      Problem 8. Last year Coca-Cola and PepsiCo had a combined revenue of $66 billion dollars.
                          If Coca-Cola received $14 billion more in revenue than PepsiCo last year, then what was the revenue
                          for each company last year?

Solving word problems using linear systems of two equations in two unknowns

      Problem 1. The Madison Local High School marching band sold gift wrap to earn money for a band trip to Orlando, Florida.
                          The gift wrap in solid colors sold for $4.00 per roll, and the print gift wrap sold for $6.00 per roll.
                          The total number of rolls sold was 480, and the total amount of money collected was $2340.
                          How many rolls of each kind of gift wrap were sold?

      Problem 2. If 4 apples and 2 oranges cost $1 and 2 apples and 3 orange cost $0.70, how much does each apple and each orange cost?
                          There are no quantity discounts.

      Problem 3. Tickets are sold at $4.00 for adults and $2.50 for students. If 100 tickets were sold for $355.00,
                          how many tickets were adult tickets?

      Problem 4. A piece of metal is 1 ft. long, 6 in. wide, and 4 in. thick, and weighs 189.8125 pounds.
                          It is composed of an alloy of gold and copper. Determine percentage of gold.
                          Gold density is 0.70 lbs. per cu. in.
                          Copper density is 0.32 lbs. per cu. in.

Oranges and grapefruits

      Problem 1. On Monday you bought 11 oranges and 4 grapefruit for a total of $7.40. Later that day, you bought 3 oranges and 5 grapefruits
                          for a total of $4.95. Find the price of each orange and each grapefruit.

Using systems of equations to solve problems on tickets

      Problem 1. Adult tickets for a concert cost $8 each and student tickets cost $5 each. A total of 55 tickets were sold worth $365.
                          How many adult and student tickets were sold?

      Problem 2. A movie theater charges $5 admission for an adult and $3 for a child. If 700 tickets were sold
                          and the total revenue received was $2900, how many tickets of each type were sold?

      Problem 3. If 104 people attend a concert and tickets for adults cost $2.255 while tickets for children cost $1.75
                          and total receipts for the concert was $202.5, how many of each went to the concert?

Three methods for solving standard (typical) problems on tickets

      Problem 1. 78 people attended the amusement park and the cost was $444. If adults costs $8
                          and students costs $5, how many adults and students attended ?

                          The three methods are: 1) Two equation approach;   2) One equation approach,   and 3) Logical analysis without using equations (MENTAL solution).

Using systems of equations to solve problems on shares

      Problem 1. Mary spends $2,860 to buy stock in two companies. She pays $13 a share to one of the companies and $26 a share to the other.
                          If she ends up with a total of 150 shares, how many shares did she buy at $13 a share and how many did she buy at $26 a share?

Using systems of equations to solve problems on investment

      Problem 1. Madison invested a total of $30,000 at two different banks. At one bank she earned 3.5% on her investment
                          and at the other bank she earned 4.5%. If her total earning per year is $1,320 then how much did she invest at each bank?

      Problem 2. Sue has $80,000 to invest in a savings account, which pays 7% and a certificate of deposit which pays 8.4%.
                          Sue would like to receive $6,300 as interest income. How much should she invest in each?

      Problem 3. $15,074 is invested part at 14% and the rest at 5%. If the interest earned from the amount invested at 14% exceeds
                          the interest earned from the amount invested at 5% by $1694.07, how much is invested at each rate?

      Problem 4. Jack inherited 250000 pesos and invested money in SM, Meralco, and Manila Water. After a year, he got a small return of 16200 pesos
                          from the three investments. SM returned 6%, Meralco returned 7%, and Manila Water returned 8%. There was 60000 more pesos invested
                          in Meralco than in Manila Water. How much did he invest in SM, Meralco, and Manila Water?

      Problem 5. Joseph invested a total of $10,000 in two accounts. After a year, one account lost 7.8%,
                          while the other account gained 6.7%. In total, Joseph lost $562.50.
                          How much money did Joseph invest in each account?

Two mechanics work on a car

      Problem 1. Two mechanics worked on a car. The first mechanic worked for 20 hours, and the second mechanic worked for 15 hours.
                          Together they charged a total of $2250.
                          What was the hourly rate charged per hour by each mechanic, if the sum of the two rates was $125 per hour?

The Robinson family and the Sanders family each used their sprinklers last summer

      Problem 1. The Robinson family and the Sanders family each used their sprinklers last summer.
                          The water output rate for the Robinson family's sprinkler was 25 L per hour.
                          The water output rate for the Sanders family's sprinkler was 20 L per hour.
                          The families used their sprinklers for a combined total of 55 hours, resulting in a total water output of 1250 L.
                          How long was each sprinkler used?

      Problem 2. The Johnson family and the Taylor family each used their sprinklers last summer.
                          The Johnson family' sprinkler was used for 20 hours. The Taylor family' sprinkler was used for 30 hours.
                          There was a combined total output of 1450 L of water.
                          What was the water output rate for each family' sprinklers if the sum of the two rates was 55 L per hour?

Roses and violets

      Problem 1. Kaiden has sold 10 bunches of roses and 12 violets for a total of $380. Grayson has sold 6 bunches of roses and 8 violets for $244.
                          What is the cost of a bunch of roses? What is the cost of a bunch of violets?

Counting calories and grams of fat in combined food

      Problem 1. Two foods are combined to give a total of 1600 calories and 17 g of fat. The first food has 200 calories per ounce and 3 g of fat per ounce,
                          the second food has 250 calories per ounce and 2 g of fat per ounce. How many ounces of each food is used?

A theater group made appearances in two cities

      Problem 1. A theater group made appearances in two cities. The hotel charge before tax in the second city was $1500 higher than in the first.
                          The tax in the first city was 6.5%, and the tax in the second city was 3.5%.
                          The total hotel tax paid for the two cities was $427.50. How much was the hotel charge in each city before tax?

      Problem 2. A theater group made appearances in two cities. The hotel charge before tax in the second city was $500 lower than in the first.
                          The tax in the first city was 6.5%, and the tax in the second city was 9.5%. The total hotel tax paid for the two cities was $512.50.
                          How much was the hotel charge in each city before tax?

Exchange problems solved using systems of linear equations

      Problem 1. Arthur and Sam shared some cards. If Arthur gave 20 cards to Sam, they would have the same number of cards.
                          If Sam gave Arthur 10 cards, the ratio of cards Arthur had to Sam is 2 to 1. How many cards did Arthur have ?

      Problem 2. Sam and Lynn had 96 storybooks altogether. Sam gave 1/4 of his storybooks to Lynn.
                          Lynn then gave 1/3 of the total number of storybook she had to Sam.
                          In the end, they had the same number of storybooks. How many storybooks did each of them have at first?

Typical word problems on systems of 2 equations in 2 unknowns

      Problem 1. Anjali and Joe improved their yards by planting daylilies and ivy. They bought their supplies from the same store.
                          Anjali spent $233 on 13 daylilies and 14 pots of ivy. Joe spent $109 on 5 daylilies and 7 pots of ivy.
                          Find the cost of one daylily and the cost of one pot of ivy.

      Problem 2. A merchant mixed 12 lb of a cinnamon tea with 5 lb of spice tea. The 17-pound mixture cost $28.
                          A second mixture included 14 lb of the cinnamon tea and 8 lb of the spice tea. The 22-pound mixture cost $37.
                          Find the cost per pound of the cinnamon tea and of the spice tea.

      Problem 3. A carpenter purchased 70 ft of redwood and 80 ft of pine for a total cost of $335.
                          A second purchase, at the same prices, included 100 ft of redwood and 50 ft of pine for a total cost of $395.
                          Find the cost per foot of redwood and of pine.

      Problem 4. The total calories gained from eating 2 oranges and 3 strawberries is 112 calories.
                          On the other hand, the total calories gained from eating 3 oranges and 4 strawberries is 165
                          calories. How much calories does one orange contribute and how much calories does one strawberry contribute?

      Problem 5. Jesse and Adam are both selling items to raise money for baseball uniforms.
                          They are each selling two items: chocolate bars and candles. Jesse sells 12 chocolate bars and 7 candles
                          and makes a total of $94. Adam sells 24 chocolate bars and 5 candles and makes a total of $98.
                          How much does one chocolate bar and one candle cost?

      Problem 6. A small tool manufacturer produces a standard and cordless model power drill.
                          The standard model takes 2 hours of labor to assemble and the cordless model 3 hours.
                          There are 72 hours of labor available per week for the drills.
                          Material costs for the standard drill are $10, and for the cordless drill they are $20.
                          To maximize profits, the company wished to limit material costs to $420 each week.
                          How many of the cordless drills should be produced to use all of the available resources?

      Problem 7. Sheena’s Bakery makes and sells cookies and cakes.
                          It costs the store Php580 to buy the supplies needed to make 20 cookies, and Php400 for
                          the supplies needed to make a cake. The store sells the cookies for Php40 each and the cakes for Php620 each.
                          Last month Sheena's Bakery spent Php9880 on supplies and sold all of the cookies and cakes that were made last month
                          using those supplies for Php14,720. How many cookies and cakes did they make?

      Problem 8. If Bob sells a TV at a discount of 15% of the marked price, he will make a profit of $135.
                          If he sells at a discount of 25% of the marked price, he makes a loss of $65.
                          What is the buying price, at which Bob buys TVs from the TV's producer?

HOW TO algebraize and solve these problems on 2 equations in 2 unknowns

      Problem 1. A stadium has 50,000 seats. Seats sell for $25 in section A, $20 in section B, $15 in section C.
                          The number of seats in section A equals the total number of seats in sections B and C.
                          Suppose the stadium takes in $1,074,500 from each sold-out event. How many seats does each section holds?

      Problem 2. Bruce collects stamps. He has six times as many 10-cent stamps as 5-cent stamps, and he has some 20 cent stamps as well.
                          Bruce has 72 stamps with a total value of $8.40. How many of each stamp does he have?

      Problem 3. Sally is going to buy a total of 11 new items at Target. She is going to buy jeans, dresses, and shoes.
                          She is going to spend exactly $460 and has discovered that jeans are $25, dresses are $50, and shoes are $40.
                          She is also going to buy twice as many shoes as jeans. Find out how many jeans, how many shoes, and how many dresses she will buy?

      Problem 4. Jack and Jill spent two weeks touring Boston, New York City, Philadelphia, and Washington D.C.
                          They paid $120, $200, $80, and $100 per night respectively. Their total bill was $2020.
                          The number of days spent in NYC was the same as the sum of the days spent in Boston and D.C.
                          They spent three times as many days in NYC as they did in Philly. How many days did they stay in each city?

      Problem 5. Two cars, A and B, 148 kilometers apart, traveled toward each other. A started at 6 a.m. while B started an hour later.
                          They met on the road at 9 a.m. Had both of them started at 6 a.m., they would have met at 8:28. Find the speeds.

      Problem 6. John and Nat were given some money. If John spends $50 and Nat spends $100 each day, John would still have $2500 left
                          while Nat would have spent all her money. If John spends $100 and Nat spends $50 each day, John would still have $1000 left
                          while Nat would have spent all her money. How much money John and Nat were given each?

      Problem 7. At first, the ratio of Bob's and Ted's money was 5 to 2. After each of them spent an equal amount,
                          the ratio of bob to teds became 6 to 1. At the end, they had a total 126 dollars. How much did they spend together?

      Problem 8. May spent 1/6 of her money on a dress and 2 blouses. The dress costs as much as 3 blouses.
                          She spends 3/4 of the remaining money on a watch. The watch costs $220.50 more than the dress.
                          How much did she have at first.

      Problem 9. Manuel bought 6 candy bars and 3 sodas at the gas station for $8.40. Gary bought
                          3 candy bars and 4 sodas at the same gas station for $7.45. What is the price of one soda?

      Problem 10. Mrs. Lee bought 20 ducks and chickens altogether for 126 dollars. Each chicken costs 2 dollars less than each duck.
                          If she bought 6 more chickens than ducks, how much did she pay for each chicken ?

      Problem 11. Daniel has one-peso coins in the left and right pockets of his pants.
                          If he transfers one coin from his left pocket to his right pocket,
                          the number of coins in his right pocket would be twice of what he has on the left.
                          However, if he transfers one coin from his right to his left, the number of coins
                          in his pockets will be equal. How many coins does he have in his right pocket?

      Problem 12. A tortoise makes a journey in two parts; it can either walk at 4 cm/s or crawl at 3 cm/s.
                          If the tortoise walks the first part and crawls the second, it takes 110 seconds.
                          If it crawls the first part and walks the second, it takes 100 seconds.
                          Find the lengths of the two parts of the journey.

      Problem 13. The monthly salaries of two men are in the ratio 3:2
                          and their expenditures are in the ratio 8:5.
                          Each man saves 500 dollars every month. Find their monthly salaries.

      Problem 14. A farmer has chickens and ducks in his farm.
                          If he sells 2 chickens and 3 ducks a day, there would be 100 chickens left when all the ducks are sold.
                          If he sells 3 chickens and 2 ducks a day, there would be 25 chickens left after all the ducks are sold.
                          Find the number of ducks and chickens on the farm.

      Problem 15. A factory makes use of two basic machines, A and B, which turn out two different products, yarn and thread.
                          Each unit of yarn requires 1 hour on machine A and 2 hours on machine B, while each unit of thread requires 1 hour on A and 1 hour on B.
                          Machine A runs 8 hours per day, while machine B runs 14 hours per day.
                          How many units each of yarn and thread should the factory make to keep its machines running at capacity?

One unusual problem to solve using system of two equations

      Problem 1. Three kinds of tickets are sold for a school student parent dinner: a $1.00 ticket for one adult,
                          $1.50 ticket for one adult and one child and $2 ticket for 2 adults and one child.
                          Total raised was $69 and 32 children and 62 adults attended. How many of each type of ticket was sold?

Non-standard problems with a tricky setup

      Problem 1. Alex and Belinda have some money.
                          If Alex spends 40 dollars and Belinda spends 80 dollars daily Belinda will have 1260 dollars left when Alex spends all of his money.
                          If Alex spends 80 dollars and Belinda 40 dollars daily Belinda will have 1500 dollars left when Alex has spent all his money.
                          How much money do Alex and Belinda have ?

      Problem 2. Suppose a function f is such that   f(1/x) - 3f(x) = x   for every non-zero x.   Find f(2).

Sometimes one equation is enough to find two unknowns in a unique way

      Problem 1. A restaurant offered an eat-all-you-can menu last November.
                          They charged 250 pesos for kids below 3 feet and 350 pesos for adults or kids beyond 3 feet.
                          If the restaurant earned 12,450 on one day, what is the minimum number of kids below 3 feet,
                          if there are more customers who paid 350 pesos?

      Problem 2. Lyza is selling kakanin before she goes to school.
                          She sells putubumbong and bibingka at 1 0.50 pesos and 12.75 pesos per slice, respectively.
                          Her total earnings today is 389.25 pesos.
                          If she sold more putubumbong than bibingka, what is the total number of kakanin sold?

      Problem 3. A factory sorts pencils into bags such that 19 large bags plus 3 small bags contain a total of 224 pencils.
                          Find the number of pencils in a large bag and the number of pencils in a small bag.

Solving mentally word problems on two equations in two unknowns

      Problem 1. On Monday, Revlien paid 66 pesos for 3 doughnuts and 2 cups of coffee.
                          On Tuesday, she paid 69 pesos for 2 doughnuts and 3 cups of coffee.
                          How much did he pay for 1 doughnut and 1 cup of coffee on Wednesday?

      Problem 2. Four times a certain number increased by three times a second number is 25.
                          Four times the first number decreased by three times the second number is 7. Find the two numbers.

      Problem 3. Ben and Greg go to the movies and purchase snacks. Ben purchases four cookies and two bottles of water for a total cost of $9.00.
                          Greg purchases two cookies and two bottles of water for a total cost of $7.00. What is the cost of a bottle of water?

      Problem 4. An event organizer is reserving rooms for two company-wide events.
                          For the quarterly meeting this month, she reserved 3 conference rooms and 5 ballrooms, which can seat a total of 379 attendees.
                          For safety training next month, she reserved 3 conference rooms and 4 ballrooms, which can seat 314 attendees.
                          How many attendees can each room accommodate?

      Problem 5. Kira wanted to stock up on drinks for an upcoming party.
                          First she spent $53 on 14 cases of juice and 11 cases of soda, which is all the store had in stock.
                          A few days later, she returned to the store and purchased an additional 14 cases of juice and 8 cases of soda, spending a total of $50.
                          What is the price of each drink?

      Problem 6. One Friday night, two large groups of people called centerville taxi service.
                          The first group requested 3 sedans and 2 minivans, which can seat a total of 21 people.
                          The second group asked for 3 sedans and 3 minivans, which can seat a total of 27 people.
                          How many passengers can each type of taxi seat?

      Problem 7. Suppose that the weight (in pounds) of an airplane is a linear function of the amount of fuel (in gallons) in its tank.
                          When carrying 18 gallons of fuel, the airplane weighs 1999 pounds. When carrying 50 gallons of fuel, it weighs 2175 pounds.
                          How much does the airplane weigh if it is carrying 60 gallons of fuel?

      Problem 8. The remaining credit on a prepaid phone card after 22 minutes of calls is $21.70 and the remaining credit after 58 minutes of calls is $16.30.
                          What is the remaining credit after 74 minutes of calls?

      Problem 9. A merchant mixed 10 lb of a cinnamon tea with 4 lb of spice tea. The 14-pound mixture cost $37.
                          A second mixture included 16 lb of the cinnamon tea and 4 lb of the spice tea. The 20-pound mixture cost $52.
                          Find the cost per pound of the cinnamon tea and of the spice tea.

      Problem 10. Jayce bought 2 bath towels and 3 hand towels for $35. His sister jayna bought 3 bath towels and 3 hand towels for $45.
                          Determine the prices of a bath towel and hand towel.

Solving systems of non-linear equations by reducing to linear ones

      Problem 1. Solve this system of non-linear equations

                 +  = 1,

                 -  = 5.

Problem 2. Solve the system of non-linear equations

                 +  = 5,

                 +  =13.

Problem 3. Solve the system of non-linear equations

                6x + 5y +  = 72,

                3x - 4y +  = 30.

Solving word problems for 3 unknowns by reducing to equations in 2 unknowns

      Problem 1. VIP seating, reserved seating, and general admission tickets were sold
                          for the school play at $15, $10 and 5$ each respectively. The drama department
                          sold 360 tickets for a total of $2800. If there were 40 more general admission tickets
                          than the total number of VIP and reserved tickets, how many of each type of ticket were sold?

      Problem 2. A vending machine's coin box contains nickels, dimes, and quarters. The total number of coins in the box is 296.
                          The number of dimes is three times the number of nickels and quarters together. If the box contains 29 dollars and 30 cents,
                          find the number of nickels, dimes and quarters that it contains.

System of equations helps to solve a problem for the Thanksgiving day

      Problem 1. Kylie was buying pies at the store for her co-workers on Thanksgiving. She needs a total of 12 pies.
                          The store sells pumpkin pies for $4 a pie and apple pies for $2 a pie. If Kylie ended up spending $30 on pies alone,
                          how many of each pie did she purchased?

Using system of two equations to solve the problem for the day of April, 1

      Problem 1. You burn approximately 230 calories less per hour if you ride your bike versus go on a run.
                          Lien went on a 2-hour run plus burned an additional 150 calories in his warm up and cool down.
                          Theo went on a 4-Hour bike ride. Lien and Theo burned the same amount of calories on their workouts.
                          Approximately how many calories do you burn in an hour for each type of exercise?

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