1. Solving Linear Equations

2. Solving Linear Equations
Example 1
It’s obvious what the answer is. However, we need to start with the basics and work our way up because we need to make sure that we have GREAT fundamentals in math. In other words, we need to know WHY we do things other than “That’s what my teacher told me to do.”
Next question: why do we subtract the three?

3. Solving Linear Equations
Example 2
Most people know that the first thing you should do on this problem is add five to both sides. But why? Why not divide by the two first? Why not move the nine? Why is the five first?
We move the five rather than the nine because the five is on the same side of the equals sign as the variable. We move the five before we move the two because of the order of operations.
When you are solving an equation, you are undoing the problem. Therefore you use the order of operations in reverse order. Let me explain…

4. Solving Linear Equations
ORDER OF OPERATIONS P E M D A S
ARENTHESES
XPONENTS
ULTIPLICATION
IVISION
DDITION
UBTRACTION

5. Solving Linear Equations
ORDER OF OPERATIONS
PARENTHESES
EXPONENTS
MULTIPLICATION
DIVISION
ADDITION
SUBTRACTION
PEMDAS is used for evaluating expressions with no variables, like this:

6. Solving Linear Equations
ORDER OF OPERATIONS
SUBTRACTION
ADDITION
DIVISION
MULTIPLICATION
EXPONENTS
PARENTHESES
However, since we are solving an equation, we are UNDOING the order of operations. Therefore, we need to UNDO things in reverse order.

7. Solving Linear Equations
Example 2
We add five to both sides since it is subtracted from the variable.
We move the two by division because it is multiplied times the variable.

8. Solving Linear Equations
Example 2
One last thing before we move on. When I say that x = 7, I am saying TWO things. What are the two things?
The first thing that I am saying is that 7 works. That means when I plug 7 in for the x, I get a TRUE statement.

9. Solving Linear Equations
Example 2
One last thing before we move on. When I say that x = 7, I am saying TWO things. What are the two things?
What else am I saying?
I am saying that NOTHING ELSE works.

10. Solving Rational Equations
Example 3
This one is for you to try.
Can you get a fraction for an answer?

11. Solving Rational Equations
Example 4
Why is this problem different than the others we’ve seen so far?
Because there’s a variable on both sides of the equation.
Can you check your answers?

12. Solving Rational Equations
Example 4
There are three rules when checking your answers:
Always plug the answers back into the ORIGINAL problem,
Never cross the equals sign, and
Use the order of operations on each side of the equation.

13. Solving Rational Equations
Example 5
Before working this problem, get rid of the parentheses by distributing the outside numbers.
If there are like terms on the same side of the equals sign, just combine them as they are.

14. Solving Rational Equations
Example 5
Can you check these?

15. Solving Rational Equations
Example 6
Don’t forget to multiply by the number and the sign in front of the number.

16. Solving Rational Equations
Example 7
Fractions!!!
You only need to know one “trick” to solve equations with fractions. To multiply a fraction times a whole number, divide the bottom into the whole number, then multiply what’s left.
Let’s practice a couple of those before we work this equation.

17. Solving Rational Equations
Example 7
( ) 12
Back to the problem.
Find the common denominator of all of the fractions and multiply everything by that.
Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left.

18. Solving Rational Equations
Example 8
( ) 15
Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left.

19. Solving Rational Equations
Example 10
What happened that is different from the other problems?
If the variables go away, then the answer is either “no solutions” or “all real numbers”.
It’s “no solutions” if what’s leftover is a false statement.
It’s “all real numbers” if what’s leftover is a true statement. O

20. Solving Rational Equations
Example 11
Remember, if the variables go away, then the answer is either “no solutions” or “all real numbers”.
It’s “no solutions” if what’s leftover is a false statement.
It’s “all real numbers” if what’s leftover is a true statement.
ALL REAL NUMBERS

21. Solving Rational Equations
Example 12
Is this “no solutions” or “all real numbers”?
Neither. Remember, it’s only one of those if the VARIABLES go away. Just keep on working this problem.

22. Solving Rational Equations
Pair Practice: Solve these problems with a partner. 1) 2) 3) 4) 5)

23. Solving Rational Equations
Answers to worksheet #1:

24. It’s time for… HEY STUPID!!!
The Rules are very simple. Be the first person to identify the mistake made in the following problems, raise your hand and hollar “Hey, Stupid!” The first correct answer wins you five bonus points on the Chapter Test. Once you win once, you cannot win again until next chapter.
It’s time for… HEY STUPID!!!

25. Solving Rational Equations
HEY STUPID!!! #1
Didn’t distribute the 3 in the first step.

26. Solving Rational Equations
HEY STUPID!!! #2
Supposed to subtract the 5 instead of the 3.

27. Solving Rational Equations
HEY STUPID!!! #3
Should be no solutions, because the statement that is leftover is NOT true.
ALL REAL NUMBERS

28. Solving Rational Equations
HEY STUPID!!! #4
Didn’t distribute the negative in the first step.

29. Solving Rational Equations
Hey, Stupid! #5
( ) 6
You have to multiply EVERYTHING by the common denominator. They didn’t multiply the four by six.