Problem Solving Techniques:
Writing Equations

As an engineer, you will be called upon to solve new problems almost every day.  These new problems won't always have all the information given and they most likely will not be found within your textbooks.  Instead, you will have to be able to know how to write your own equations to describe the problem.  Then you'll have to know when you have enough equations to find the answer.  You will also need to know when you've written an incorrect equation that would lead to the wrong answer.  This section of material deals with the question:  How many equations do I need to solve the problem?

Essentially, you need as many equations as you have unknown variables.   If I have one
equation and one unknown, I can solve for the unknown:

                X + 2 = 3

                Rearrange to get X = 3-2 = 1.

Or I can have two equations with two unknowns:

            X + 3 Y = 4
            3 X - Y = 2

            To solve, I rearrange one of the equations to get get an unknown by itself on one side of the equals sign.  So:

            X = 4 - 3 Y.  Plug this into the second equation to get:

            3 (4 - 3 Y) - Y = 2
            12 - 9 Y - Y = 2
            -10 Y = -10
            Y = 1
        Plug this back into the rearranged equation to get:

            X = 4 - 3 = 1

            So, X = Y = 1.

We could have gone through the solution a few other ways.  We could have solved the
first equation for Y and then plugged it into the second equation.  Or, we could have solved the second equation for Y and plugged it into the first equation.  And, finally, we could
have solved the second equation for X and plugged it back into the first equation.

It doesn't matter which way we do it.  We'll get the same answer every time we solve it!
Now would be a good time to go through each of the solutions to prove to yourself that it does give you the same answer every time...

Now, let's say we had 100 equations and 100 unknowns.  Could we solve for all the unknowns?  Yes, we could.  Would it be easy?  Probably not.  This type of large problem, which often appears in process control and process design research, is best solved using linear algebra and matrices.

What happens when we have too many equations?  When this happens, our problem may be overspecified.  If we solve the problem one way, we get one answer, but if we solve it another way we would get a different answer.  Let's do an example:

            X + 3 Y = 4
            3 X - Y = 2
            X + Y = 3

If we solve the first two equations, we get that X = Y = 1.  Now plug that back into the third equation and we find that:

            1 + 1 = 3 ???!??!?

We know that one and one don't make three so we know this problem is overspecified!  Normally, one of the equations that we have written just isn't correct and contradicts one of the other equations.

Let's do another example.  Try to solve the following three equations and see what you get:

            18 X + Y = 20
            9/5 X + 1/10 Y = 2
            X - Y = -1

Try solving the first two equations by rearranging and plugging one equations into another:

            Y = 20 -18 X
        And, 9/5 X + 1/10 (20 -18 X) = 2
            1.8 X + 2 - 1.8 X = 2
           1.8 X - 1.8 X = 2 - 2
            0 = 0
So, we managed to prove that zero was equal to zero.  Nice, true, but not very useful for solving the problem.  This phenomena was caused by having two equations that were really the same equation and trying to solve the problem with those two equations.  Here, the first equation is just 10 times the second equation.  If we had tried to solve the problem with
the first and third equations, or with the second and third equations, we would have gotten the correct answer.

When one equation is just a multiple of another equation, the two equations are said to be linearly dependent.  This will lead to 0 = 0, or 1 = 1, or some other nonsensical answer.
Try to solve these equations using the first and third equations.  What was your answer?

Now try to solve it using the second and third equations.  What was your answer?

Here is an outline of how to generate and use equations to solve a problem:

1)  Write an equation.
    Make sure that the equation you wrote makes sense.
    If you have only one unknown, solve for it.  If you have more than one, go to step two.

2)  Write another equation.
    Does this one make sense?  Is it just a multiple of the first equation?  If it isn't, do you have the same number of equations as you do unknowns?  If the number of unknowns is greater than the number of equations you've written, repeat step 2.  Keep going until you have
the same number of equations as you have variables.

3)  Solve the problem.

4)  Check your answer.
    It's surprising to see how many people make mathematical errors when they are solving problems.  It's always a good idea to plug all of your unknowns into the equations again to make sure that you didn't make a mistake.

 Click to return to learn more techniques .

2007 Arizona Board of Regents for The University of Arizona