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
Rearrange to get X = 3-2 = 1.
Or I can have two equations with two unknowns:
X + 3 Y = 4
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
X = 4 - 3 = 1
So, X = Y = 1.
We could have gone through the solution a few other ways.
We could have solved the
It doesn't matter which way we do it. We'll get
the same answer every time we solve it!
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
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
Try solving the first two equations by rearranging and plugging one equations into another:
Y = 20 -18 X
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.
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.
2) Write another equation.
3) Solve the problem.
4) Check your answer.
Click to return to learn more techniques .
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