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Session 9 Part A Part B Part C Part D Part E Homework
 
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Session 9, Part D:
Working with Algebraic Structures (60 minutes)

In This Part: Units Digit Equations | Cryptography

How might you solve equations in this new system? Note 7

Let's start with the equation 3x = 8. Looking at the multiplication table, we can see that 3 * 6 = 8, so x = 6 is a solution. In fact, it's the only solution, because there is only one "8" in the third row of the multiplication table.

Or you could reason like this: "To solve 3x = 8 in our regular system, I would divide both sides by 3. That's the same as multiplying by the reciprocal of 3. In this system, 3 * 7 = 1, so 7 is the reciprocal of 3."

So you can calculate as follows:

3x = 8
7(3x) = 7 * 8
(7 * 3)x = 6 (Note that in the table, 7 * 8 = 6)
x = 6

Multiplying by 7 is the equivalent of dividing by 3 in this system. If you need to subtract, you can add the opposite of a number.

If you needed to solve the equation x + 4 = 2, you could reason like this: "To solve x + 4 = 2, I want to subtract 4 from both sides. In this system, 6 is the opposite of 4, so I can add 6 to both sides in order to remove the 4."

You would calculate as follows:

x + 4 = 2
x + (4 + 6) = 2 + 6
x + 0 = 8
x = 8

Try to solve these equations. If you have trouble, you can always use the operations table.

Problem D1

Solution  

Solve the equation 7x + 5 = 9 in this system. Explain how you did it. Note 8


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Remove the 5 first by adding its opposite, which is 5. Then "divide" by 7 by multiplying by the reciprocal of 7, which is 3.   Close Tip

 

Problem D2

Solution  

Solve the equation 3x + 7 = 4 in this system. Explain how you did it.


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
Add the opposite of 7, then multiply by the reciprocal of 3.   Close Tip

 

Problem D3

Solution  

Solve the equation 4x + 1 = 9 in this system. Explain how you did it.


Stop!  Do the above problem before you proceed.  Use the tip text to help you solve the problem if you get stuck.
This problem is much harder! Why? Because 4 does not have a reciprocal. There may be no solution, or there may even be more than one solution!   Close Tip

 

Problem D4

Solution  

Solve the equation 4x + 1 = 8 in this system. Explain how you did it.


Take it Further

Problem D5

Solution

Describe conditions that make the equation Ax = B in this system have no solution, exactly one solution, or more than one solution. How is this different from solving equations in the real number system?


 

Next > Part D (Continued): Cryptography

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