solving linear equations. example 1 it’s obvious what the answer is. however, we need to start...

Solving Linear Equations


5x

Example 1

83 x 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.”Why do we subtract the three?

3 3

Next question: why do we subtract the three?


Example 2

952 xMost 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…


ORDER OF OPERATIONS

PEMDAS

ARENTHESESXPONENTSULTIPLICATIONIVISIONDDITIONUBTRACTION


ORDER OF OPERATIONS

PARENTHESESEXPONENTSMULTIPLICATIONDIVISIONADDITIONSUBTRACTION

2)7(3)53(4 2

PEMDAS is used for evaluating expressions with no variables, like this:


ORDER OF OPERATIONS

SUBTRACTIONADDITIONDIVISIONMULTIPLICATIONEXPONENTSPARENTHESES

However, since we are solving an equation, we are UNDOING the order of operations. Therefore, we need to UNDO things in reverse order.


142 x

7x

Example 2

952 xWe 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.

55

22


142 x

7x

Example 2

952 xOne 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.

55

22

99

9514

95)7(2

952

x


142 x

7x

Example 2

952 xOne 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?

55

22

91

956

95)3(2

952

x

I am saying that NOTHING ELSE works.

Solving Rational Equations

8

3

14x

Example 3

683 x This one is for you to try.

Can you get a fraction for an answer?

8143 x

3 3


x2

153 x

5x

Example 4

9265 xx Why is this problem different than the others we’ve seen so far?Because there’s a variable on both sides of the equation.

x2963 x

6 6Can you check your answers?

3 3


9)5(26)5(5

Example 4

9265 xx There are three rules when checking your answers:

1) Always plug the answers back into the ORIGINAL problem,

2) Never cross the equals sign, and

3) Use the order of operations on each side of the equation.

910625 1919


22305123 xxx

x444 x11

Example 5

)1(2)6(5)4(3 xxx

327123 xx

32

x332412 x

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.

x3

32


)111(2)611(5)411(3

Example 5

)1(2)6(5)4(3 xxx

202545 4545

Can you check these?

)10(2)5(5)15(3


625510233 xxxx

42 x2x

Example 6

)3(2)1(5)5(2)1(3 xxxx

11375 xx

7

x31172 x Don’t forget to

multiply by the number and the sign in front of the number.

x3

7


6

512

5

320

4

Example 7

2

1

44

3

3xx 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.

6

4

38

12

9

49

10

4

18 2

11

27

11

14


x4

69 x

15x

Example 7

2

1

44

3

3xx

Back to the problem.

Remember, to multiply fractions times whole numbers, divide the bottom into the whole number, then multiply what’s left.

( )

12

9 x3 6x3x3

9 9

Find the common denominator of all of the fractions and multiply everything by that.


x5

3098 x

218 x

Example 8

52

5

3

3

xx Remember, to multiply fractions

times whole numbers, divide the bottom into the whole number, then multiply what’s left.

( )

15

9 30 x3x3x3

9 9

8

21x


183123 xx

Example 10

)6(3)4(3 xx

x31812

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.

x3

O


126126 xx

Example 11

)42(3)63(2 xx

x61212

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.

x6

ALL REAL NUMBERS


102105 xx

10 10

Example 12

)5(2)2(5 xx

x210103 x

Is this “no solutions” or “all real numbers”?

x2 Neither. Remember, it’s only one of those if the VARIABLES go away. Just keep on working this problem.03 x

3 30x


25105 x

Pair Practice: Solve these problems with a partner.

)15(2)32(5 xx

)1(2)3(4 xx)1(43)4(2)2(3 xxxx

12543 xx)5

8

9)4

7)3

8)2

3)1

:

Answers

1) 2) 3) 4) 5)


5

3)8

4

13)7

4)65

8)5

12)4

17)3

36)2

3)1

Answers to worksheet #1:

25

234)16

4

1)15

3

32)14

)13

)123

14)11

)10

3)9

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.


3355 xx

5 5

HEY STUPID!!! #1

)3(3)1(5 xx

x3352 x

x3 Didn’t distribute the 3 in the first step.

82 x2 2

4x


352 x

HEY STUPID!!! #2

33Supposed to subtract the 5 instead of the 3.22 x

2 21x


915515 xx

HEY STUPID!!! #3

)35(3)13(5 xx

x1595

x15 Should be no solutions, because the statement that is leftover is NOT true.ALL REAL NUMBERS


1144 xx

4 4

HEY STUPID!!! #4

)1(2)3(3)1(4 xxx

x1143 x

x Didn’t distribute the negative in the first step.

153 x3 3

5x

229344 xxx


x2

495 x

55 x

Hey, Stupid!#5

24

2

3

3

xx You have to multiply

EVERYTHING by the common denominator. They didn’t multiply the four by six.

( )

6

9 4 x3x3x3

9 9

1x

solving linear equations. example 1 it’s obvious what the answer is. however, we need to start...

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