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Algebra

February 10, 2015

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Systems of Three Variables

We have already seen a system of two equations used to solve two variables. Now we are going to examine a system of three variables which requires three equations.

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Systems of Three Equations

Whenever you add another variable to the mix you MUST add another equation in order

to solve it.

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Systems of Three Equations: Process

We start the process of solving three equations very much like solving a system of two equations by substitution.

1.We write one equation so that it gives us a scenario similar to the following:

…

…

OR

…

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Systems of Three Equations: Process

2. We then use this information and plug it into the other two equations in place of the variable.

3. We then use these two new equations and proceed to work through them the same as we would with other two variable, two equation systems.

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Example 1

First, solve the 2nd equation for :

Then, use this information to plug into the 1st and 3rd

equations.

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Example 1-Continued

So now we have:

We simplify this:

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Example 1- Continued

We then can work with these two equations to solve for and .

We can multiply the bottom equation by 3:

This then gives us the system:

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Example 1-Continued

So then we work with this system:The ’s cancel out. Leaving us with: We then simplify this: So now we know that and .

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Example 1- Continued

We can plug this information into any of our equations from the beginning and simplify and solve for .

Using and, plug into the original equation to solve for .

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Example 1- Continued

Therefore, we know that andAll we have to do is write our solution point as

follows:

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Example 2

Because none of the equations are already simplified for one variable we must choose one to work with. The 1st equation has a lone positive that is easy to work with, so we will start there.

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Example 2- Continued

Add and Subtract to the other side:

We now will substitute in this information into the

2nd and 3rd equation we were given.

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Example 2- Continued

We then simplify both of these equations:

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Example 2- Continued

We then work with these two equations to solve for either or .

The lowest common multiple (LCM) of 26 and 16 is 208. Therefore, we will multiply the top equation by 8 and the bottom equation by 13, in order to cancel out the ’s.

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Example 2- Continued

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Example 2- Continued

We now use this information about to solve for from one of the previous equations. Choose: .

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Example 2-Continued

We now know that and . So we can use this information to solve for from one of the original three equations or the re-written equation that we worked with at the beginning.

I would choose the re-written equation:

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Example 2- Continued

Thus, we now know that and.So our solution is,

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Homework

Assignment 7-4