Real-world examples of systems of linear equations?

My sojourn in the world of 8th grade math continues. As pointless and repetitive as the exercises are, the feeble attempts by the textbook authors to make the problems relevant are worse. Here’s a “real world” example of linear equations:

  1. You and your friend together sell 58 tickets to a raffle.
  2. You sold 14 more tickets than your friend.
  3. How many did you and your friend each sell?

Even if an 8th grader were willing to consider a world in which this kind of partial information were available (maybe Enron did accounting like this?) and someone cared about the answer, why would it matter?

Readers: What’s an example of a system of linear equations that has obvious practical value, as perceived by an 8th grader?

19 Comments

  1. Yossi Kreinin

    January 19, 2017 @ 1:42 pm

    1

    Oh, that’s REALLY important! Why, if they gave me such examples, I might have actually studied math properly.

    I think math in general is useless outside science and engineering. So the question becomes what sort of scientific question or engineering problem is interesting for an 8th grader. How about determining whether a shot in a 3D game hits or misses a player, and where it hits them?

  2. PN

    January 19, 2017 @ 1:49 pm

    2

    Philg: surely there is some way we can construct a system of equations which describes the benefits obtained (or paid?) by having either three children with one dentist, one children with each of three dentists, or some mix of children, dentists, and abortions for sale? Perhaps in different venues?

  3. J. Peterson

    January 19, 2017 @ 1:56 pm

    3

    Here’s one from the bad old days of land line phones:

    Long distance fees to Europe were outrageous, like $1.50/min. BUT, the phone company will cut you a deal: for $5/month fee, they’ll lower the rate to $0.12/minute.

    So, if you want to talk to your friend in Europe, how many minutes a year do you need to talk for the phone company’s deal to work in your favor?

  4. Lord Salisbury

    January 19, 2017 @ 2:06 pm

    4

    The profit on dealing crack is $55 per sale but you need to buy a black
    market gun at $400 so you don’t get robbed. Selling weed is $15 per
    sale but it’s in better neighborhoods so you can get by without the
    gun. How many sales does it take before crack profits exceed the same
    number of weed sales?

  5. EvilKiru

    January 19, 2017 @ 2:13 pm

    5

    Attempts by adults to determine anything that has obvious practical value to an 8th grader are both hilarious and futile.

  6. wally

    January 19, 2017 @ 2:17 pm

    6

    The shoot-em-up Xbox game you and your sister wants is $75. If you contribute $20 more towards it’s purchase than your sister, how much do each of you contribute?

    Part B) Assuming you own the game equally, how much money does your sister owe you?

    Part C) Assuming you can safely ignore usury laws, what interest rate will you charge her on the money owed to you?

    Part D) Assuming she doesn’t pay you back for two years, what is the total amount owed to you at the end of the second year?

  7. Demetri

    January 19, 2017 @ 2:38 pm

    7

    My 8th grade daughter is also currently going through these same endless exercises. I think that they are only useful in that they show kids that math can model some real world situation with an actual solution. The fact that they only have simple algebra at their disposal does stunt the types of problems that can be tackled. I just wish that the curriculum didn’t drag on so much on each topic.

  8. superMike

    January 19, 2017 @ 2:55 pm

    8

    How about sports? A is running with the football at some speed at some position and b is at a different speed and position trying to tackle him, at what position on the field will a and b collide? I’m sure there’s some good stuff in aviation, too.

  9. philg

    January 19, 2017 @ 3:02 pm

    9

    Thanks for the examples, folks, but I don’t see any actual equations in your comments! How about an example with both the scenario and the corresponding set of equations?

  10. jerry

    January 19, 2017 @ 3:28 pm

    10

    You and your friend to 58 reblogs about Zayn’s leaving One Direction.
    Your friend got 14 more reblogs than you did.
    What privileges does your friend have?

  11. wally

    January 19, 2017 @ 3:38 pm

    11

    @phil, isn’t the goal to get them to set up the equations? That’s the ‘magic’ for problems like the one you posed, right? You set it up correctly, and the answer just about falls out.

    Or were you trying to see who among your readers could solve systems of linear equations? Or who was older than the average 8th grader?

  12. philg

    January 19, 2017 @ 4:14 pm

    12

    wally: I just want to be sure that the example actually does lend itself to two equations in two variables! With some of the examples above it is unclear.

  13. John R. Kleim

    January 19, 2017 @ 4:36 pm

    13

    @phil,

    I recommend that you take a look at V.I. Arnold’s

    http://jnsilva.ludicum.org/HMR13_14/Arnold_en.pdf

    For some inspiration.

    John

  14. John R. Kleim

    January 19, 2017 @ 4:40 pm

    14

    @phil:

    I suggest problem 13, even though it isn’t about systems of linear equations (some of the other problems do involve them). The way we teach math is a travesty. There is absolutely no way to inspire curiousity in kids if one follows the standard curriculum in the US.

  15. Liudvikas Bukys

    January 19, 2017 @ 5:12 pm

    15

    A cheetah runs steadily at C mph. You run steadily at H mph. Safety is 5 miles away. You have a M-mile head start. When does the cheetah catch up to you (before or after you reach safety)?

  16. John R. Klein

    January 19, 2017 @ 8:23 pm

    16

    #2 from Arnold’s problems is a system of two linear equations with two unknowns:

    “A bottle with a cork costs 10 kopecks, while the bottle itself is 9 kopecks
    more expensive than the cork. How much does the bottle without the cork
    cost?”

  17. presidentpicker

    January 19, 2017 @ 11:39 pm

    17

    I like the drug dealer example

    55x-400 > 15x
    40x-400 > 0
    x > 10

    This crime scene needs another unknown to make it a system of equations.

  18. Ted

    January 20, 2017 @ 3:10 pm

    18

    Given the sum of two numbers and the difference between those two numbers, what are the two numbers?

    True, there isn’t a practical value to this problem, but a system of simultaneous linear equations is one way to solve it. There supposedly are other ways, but that’s what immediately came to mind when I saw the problem.

  19. Neal

    January 20, 2017 @ 5:36 pm

    19

    When athletes run drills using old auto tires nobody complains that there won’t be any tires on the field during the game.

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