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Understanding the Substitutionary Aspect of a Variable

A variable such as x is first off a symbol and second off a Substitution for a Number or an Equation. So given a problem:

3x + 2xy = 16

Substituting x with x = 2 the equation becomes

3(2) + 2(2)y = 16 We can solve the equation here.

However, x can also represent something else, an entirely different expression for example. Take for instance x = 2zy Substituting into our original equation:

3(2zy) + 2(2zy)y = 16

These are both easy problems to simplify and solve. But that's not the point of this lesson. Remember, math becomes increasingly difficult and transcends just math. You might use it in Chemistry or Physics and the numbers and symbols that you use are still variable but require different notation. Take for instance this problem:

xy/z The expression x times y divided by z

Let's let x = [HAc] (Concentration of Acetic Acid) and y = [H3O+] (Concentration Hydronium) and z = 3M (3 molar). What in the world!? All this is meant to do is get you to understand that x, y and z are just substitutions for whatever we may need them to be. So substitute back into the equation and see what you get.

xy/z = The quantity [HAc]times[H30+] divided by 3M = ([HAc][H30+])/3M.

Simple. I can make a variable x equal to whatever I need it to be.

I can also define x in two different ways. For instance, x = 2y and x = 3z. What does 3x equal 1) in terms of y 2) in terms of z?

1) x = 2y Use this expression since I want everything in terms of y.
3x = 3(2)y Multiply the equation by 3 to obtain 3x
3x = 6y Simplify

2) x = 3z Use this expression since I want everything in terms of z.
3x = 3(3)z Multiply x by 3 is the same as multiplying the entire equation by 3.
3x = 9z Simplify

All this substituting, why is it important? Because sometimes you need to substitute an expression into a variable to simplify a problem. Take a look at these expressions:

3SinxCosy + 4SinyCosx = 5 (Sine and Cosine are just trigonometric values. Don't worry about them, just see if you can see a pattern)
4SinxCosy + 8SinyCosx = 7

Did you notice that both equations have a SinxCosy value? and a SinyCosx value? Instead of writing all of that mess and trying to solve, why don't we substitute these vales with A and B?

A = SinxCosy
B = SinyCosx Substituting back into our equations we get

3A + 4B = 5 and
4A + 8B = 7

See how much cleaner that is?

Keep in mind that I can substitute an expression using a variable to define two related objects. Take for instance the age of a brother or sister. Let's say that your brother Aaron is 5 times your age (Mark). So, let's substitute a siblings name with a variable and then define the other sibling using the same variable.

Mark's Age = x
Aaron's Age = y = 5x = 5 times Mark's Age

Hope this helps!

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