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Often, real-world problems aren’t very neat. There may very well be more than one piece of information you don’t know and would like to find. Setting up a system of equations allows you to solve for as many variables as you have different equations. This can be especially useful if you are trying to compare scenarios that you have written as equations. The problems below are examples of times when you might use systems of equations to compare outcomes or find information. Most are modern examples, but the last question in this set is from a 9th century math text!
You fly to visit your family multiple times per year. Your airline rewards program gives your points to spend on things like ticket discounts and seat upgrades. Your rewards structure is expressed by the following equation:
\( 3*x_1 + 10*y = \) total points
where \(x_1\) = the miles you fly in a year, and y = the amount of money you spend with your airline-sponsored credit card.
You get a notice in the mail saying that your airline has changed its rewards program for the upcoming year. The new reward structure is expressed by the following equation:
\(2x_2 + 20*y = \) total points
Where \(x_2\) = the miles you can fly in the upcoming year and y = the amount of money you will spend with your airline-sponsored credit card.
You want to figure out how the new plan will affect your rewards points. You spent $2,000 to fly 10,000 miles last year and plan on spending $2,000 again this year. How many miles would you need to fly this year to achieve the same number of points as last year?
At your current job, you work 45 hours per week. You earn $25 per hour up to 40 hours and $50 per hour for the remaining 5 hours, which are overtime.
You see an ad for a new job that pays $40 per hour. How many whole hours would you have to work per week at the new job for you to make more than you do in a week at your current job?
You want to build a bookcase with 3 shelves and a top in the following shape using 42 ft of wood that your friend gave you:
You have decided that the bookcase will look best if it is 1.5 times as tall (x) as it is wide (y). What is the widest you can make each shelf (y) and use all of the wood? Write your answer in feet.
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This past weekend, you hosted a classic movie festival for your town at the local cinema. Children aged 12 and under gained free entry to any movie for the whole weekend. People aged 13-40 paid $7 per movie ticket, and people 40 and over paid $3.50. On Saturday, you sold 150 tickets in total (including children) and allowed 13 free entries. On Sunday, you sold 100 tickets in total (including children) and allowed 6 free entries. If you grossed $714 on Saturday and $483 on Sunday, did more adults or seniors attend the film festival?
The total energy of a system is the sum of its kinetic and potential energy:
E = T + U
where E is total energy, T is kinetic energy, and U is potential energy.
The total energy of a roller coaster car at the top of a hill is \(1.458*10^6 \) Joules (1 J = \( 1\frac{kg * m^2}{s^2} \)). At this point, the roller coaster car has no kinetic energy, meaning that all of its energy is potential energy. The roller coaster car then runs down the hill and enters a loop-the-loop. At the top of the loop, half of the roller coaster car’s energy is kinetic and the other half is potential. If the roller coaster car has a mass of 4500 kg, and kinetic energy is given as \( \frac{1}{2} m * v^2 \), what is the velocity of the car at the top of the loop?
Master Question: An airport has two straight runways that intersect in one place as shown below.
Between the hours of 8:00 AM and 8:00 PM, the frequency of planes crossing the intersection point per hour on runway A can be expressed as \(x+4=y \). The frequency of planes crossing the intersection point per hour on runway B can be expressed as \(x^2+y^2=n \). What value must n be less than such that you can guarantee no two planes will cross this point at the same time and collide?
The following question was written by the ancient Indian mathematician Mahavira around 850 A.D. Mahavira is one of the mathematicians in history to independently formulate the area of a triangle and approximate square roots, among other contributions.
Master Question: Three merchants find a purse lying in the road. One merchant says “If I keep the purse, I shall have twice as much money as the two of you together.” “Give me the purse and I shall have three times as much as the two of you together” said the second merchant. The third merchant said “I shall be much better off that either of you if I keep the purse I shall have 5 times as much as the 2 of you together.” If there are 60 coins (of equal value) in the purse, how much money does each merchant have? (From Mahavira)
Write your answer in the form: a, b, c where a, b, and c are the value, in coins, merchants 1, 2, and 3 have, respectively. (ex: 10, 20, 50)