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Forum: If Only I Had a System…

Applications of Systems of Linear Equalities

The Problem:

When students are surveyed about what makes a good math Forum, at least half of the responses involve

 "discussing how to work problems"  "seeing how this math applies to real-life situations"

This Forum on applications of systems of equations addresses both of these concerns.

Unfortunately, the typical postings are far from ideal.

This is an attempt to rectify the situation. Please read this in its entirety before you post your answer!

Pick-up games in the park vs. the NBA:

Shooting hoops in the park may be lots of fun, but it scarcely qualifies as the precision play of a well-coached team. On the one hand, you have individuals with different approaches and different skill levels, "doing their own thing" within the general rules of the game. On the other hand you have trained individuals, using proven strategies and basing their moves on fundamentals that have been practiced until they are second nature.

The purpose of learning algebra is to change a natural, undisciplined approach to individual problem solving into an organized, well-rehearsed system that will work on many different problems. Just like early morning practice, this might not always be pleasant; just like Michael Jordan, if you put in the time learning how to do it correctly, you will score big-time in the end.

But my brain just doesn't work that way. . .

Nonsense! This has nothing to do with how your brain works. This is a matter of learning to read carefully, to extract data from the given situation and to apply a mathematical system to the data in order to obtain a desired answer. Anyone can learn to do this. It is just a matter of following the system; much like making cookies is a matter of following a recipe.

"Pick-up Game" Math

It is appalling how many responses involve plugging in numbers until it works.

 "My birthday is the eleventh, so I always start with 11 and work from there."  "The story involved both cats and dogs so I took one of the numbers, divided by

2 and then I experimented."  "First I fire up Excel..."  "I know in real-life that hot dogs cost more than Coke, so I crossed my fingers

and started with $0.50 for the Coke..."

The reason these "problem-solving" boards are moderated is so that these creative souls don’t get everyone else confused!

NBA Math

In more involved problems, where the answer might come out to be something irrational, like the square root of three, you are not likely to just randomly guess the correct answer to plug it in. To find that kind of answer by an iterative process (plugging and adjusting; plugging and adjusting; ...) would take lots of tedious work or a computer. Algebra gives you a relative painless way of achieving your objective without wearing your pencil to the nub.

The reason that all of the homework has involved x's and y's and two equations, is that we are going to solve these problems that way. Each of these problems is a story about two things, so every one of these is going to have an x and a y. In some problems, it’s helpful to use different letters, to help keep straight what the variables stand for. For example, let L = the length of the rectangle and W = the width.

The biggest advantage to this method is that when you have found that w = 3 you are more likely to notice that you still haven’t answered the question, “What is the length of the rectangle?” Here are the steps to the solution process:

 Figure out from the story what those two things are.

o one of these will be x

o the other will be y

 The first sentence of your solution will be "Let x = " (or “Let L = ” )

o Unless it is your express purpose to drive your instructor right over the edge, make sure that your very first word is "Let"

 The second sentence of your solution will be "Let y = " (or “Let W = ” )

 Each story gives two different relationships between the two things.

o Use one of those relationships to write your first equation.

o Use the second relationship to write the second equation.

 Now demonstrate how to solve the system of two equations. You will be using either

o substitution

o or elimination - just like in the homework.

More examples…

For this problem, I'd use substitution to solve the system of equations:

The length of a rectangle blah, blah, blah...

Let L = the length of the rectangle

... blah, blah, blah twice the width

Let W = the width of the rectangle

The length is 6 inches less than twice the width

L = 2W - 6

The perimeter of the rectangle is 56

2L + 2W =56

For this one, I'd use elimination to solve the system of equations:

Blah, blah, blah bought 2 cokes...

Let x = the price of a coke

.. blah, blah, blah 4 hot dogs

Let y = the price of a hot dog

2 cokes plus 4 hot dogs cost 8.00

2x + 4y = 8.00

3 cokes plus 2 hot dogs cost 8.00

3x + 2y = 8.00

For this one, I'd use substitution to solve the system of equations:

One number is blah, blah, blah...

Let x = the first number

...blah, blah, blah triple the second number

Let y = the second number

The first number is triple the second

x = 3y

The sum of the numbers is 24

x + y = 24

Checking your answers vs. Solving the problem

The problem: Two numbers add to give 4 and subtract to give 2. Find the numbers.

Solving the problem:

Let x = the first number

Let y = the second number

Two numbers add to give 4: x + y = 4

Two numbers subtract to give 2: x - y = 2

Our two equations are: x + y = 4 x - y = 2 Adding the equations we get

2x = 6

x = 3 The first number is 3.

x + y = 4 Substituting that answer into equation 1

3 + y = 4

y = 1 The second number is 1.

Checking the answers:

Two numbers add to give 4: 3 + 1 = 4

The two numbers subtract to give 2: 3 - 1 = 2

Do NOT demonstrate how to check the answers that are provided and call that demonstrating how to solve the problem!

Formulas vs. Solving equations

Formulas express standard relationships between measurements of things in the real world and are probably the mathematical tools that are used most frequently in real-life situations.

Solving equations involves getting an answer to a specific problem, sometimes based on real-world data, and sometimes not. In the process of solving a problem, you may need to apply a formula. As a member of modern society, it is assumed that you know certain common formulas such as the area of a square or the perimeter of a rectangle. If you are unsure about a formula, just Google it. Chances are excellent it will be in one of the first few hits.

If you are still baffled:

 Check out all the examples worked out in the PowerPoints in the Other Resources section of the Handy Helpers for Section 4.3.

 Message me if you are still confused.