variables on both sides

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Notes and examples 3-5Notes and examples 3-5

What you have learned so far What you have learned so far (about solving equations)(about solving equations)

HowHow to to solvesolve anan equationequation in in oneone stepstep by by additionaddition oror subtractionsubtraction

How How toto solve solve an an equation equation inin one one stepstep by by multiplication multiplication or or divisiondivision

HowHow to to solvesolve an an equationequation in in twotwo or or moremore steps steps by by all all methodsmethods listed listed aboveabove

What What elseelse do do youyou need need toto know know about about equationsequations??????

A whole A whole bunch !!!bunch !!!

TheThe rules rules araree sstilltill the same the same…..…..

Solve: 12x + 21 = 9xSolve: 12x + 21 = 9x

You have to get the x’s all on the same side. To do this, you You have to get the x’s all on the same side. To do this, you move themove the 12x AWAY from the side with the 21 by 12x AWAY from the side with the 21 by subtracting it from both sides…..subtracting it from both sides…..

12x + 21 = 9x12x + 21 = 9x

- 12x- 12x -12x-12x

21 = - 3x21 = - 3x

Now you have a one step equation to solve: divide both sides Now you have a one step equation to solve: divide both sides by – 3:by – 3:

2121 = = - 3x- 3x

- 3 - 3- 3 - 3

-7 = x-7 = x

Let’sLet’s try try ananother other one!one!4n = -28n – 34n = -28n – 3

This time, the variable on the This time, the variable on the LEFT needs to move….LEFT needs to move….

4n = - 28n – 324n = - 28n – 32

+28n+28n +28n+28n

32n = - 32 32n = - 32

Divide both sides by 32:Divide both sides by 32:

32n32n = = - 32- 32

32 3232 32

n = - 1n = - 1

AA little little bit bit harderharder…..…..

8a - 4(-5a - 2) = 12a8a - 4(-5a - 2) = 12a11stst, do the distributive property to get , do the distributive property to get rid of the parentheses:rid of the parentheses:

8a + 20a + 8 = 12a8a + 20a + 8 = 12a

Add like terms on each side:Add like terms on each side:

28a + 8 = 12a28a + 8 = 12a

Now it looks like the other ones that Now it looks like the other ones that we have solved!!we have solved!!

To To solvesolve for for a,a, now now youyou need need to to subtract subtract 28a28a from from eacheach side: side:

28a + 8 = 12a28a + 8 = 12a

-28a-28a -28a-28a

8 = -16a8 = -16a

And solve as usual! (divide both sides by And solve as usual! (divide both sides by what???)what???)

88 = = -16a-16a

-16 -16-16 -16

-1/2 = a-1/2 = a

TTry try thihis ones one!!

––5(6 - 5n) = 10(2n - 6)5(6 - 5n) = 10(2n - 6)

What What would would you you dodo FIRST FIRST ????

IIf f yyou ou ssaid aid tthe he DDistributive istributive PPropertyroperty….….

YoYou’u’re Rire Righght!!t!!

Here’s how it should look:Here’s how it should look:

––5(6 - 5n) = 10(2n - 6)5(6 - 5n) = 10(2n - 6)

-30 + 25n = 20n – 60-30 + 25n = 20n – 60

Now, you have to move the variables Now, you have to move the variables together on one side, and the numberstogether on one side, and the numbers without variables to the other side!without variables to the other side!

First, you could add 30 to each side……First, you could add 30 to each side……

-30 -30 ++ 25n = 20n 25n = 20n – – 6060 +30+30 +30+30Make sure you only add or subtract numbers Make sure you only add or subtract numbers

with other numbers, and letters with other with other numbers, and letters with other letters (like terms): Now we haveletters (like terms): Now we have

25n = 20n – 3025n = 20n – 30Subtract: Subtract: -20n-20n -20n-20n - 30 - 30

Now we have Now we have 5n = -305n = -30

Divide both sides by 5, to get n = - 6Divide both sides by 5, to get n = - 6

Pretty Easy, huh? (just kidding!)Pretty Easy, huh? (just kidding!)

OK, here’s another OK, here’s another one to try:one to try:

½(12n - 4) = 14 - ½(12n - 4) = 14 - 10n10n

( Without any help ( Without any help from me!! )from me!! )

How did you do?How did you do? ½(12n - 4) = 14 – 10n½(12n - 4) = 14 – 10n 6n - 2 = 14 – 10n6n - 2 = 14 – 10n +10n+10n + 10n+ 10n 16n - 2 = 1416n - 2 = 14 + 2+ 2 +2+2 16n16n = = 1616 16 1616 16 n = 1n = 1

One more thing…..One more thing…..

When you are working these problems….it is When you are working these problems….it is possible to end up with a FALSE possible to end up with a FALSE statement: such as--statement: such as--

1. The variables cancel each other out, and 1. The variables cancel each other out, and you are left with just numbers, like 5 = 13, you are left with just numbers, like 5 = 13, which CAN NOT be true – so the answer is:which CAN NOT be true – so the answer is:

NO SOLUTION!NO SOLUTION!2. The numbers cancel each other out and 2. The numbers cancel each other out and

you are left with just variables, like 2x = you are left with just variables, like 2x = 3x: which CAN be true, only if x = 0! So 3x: which CAN be true, only if x = 0! So your answer for that kind is your answer for that kind is x = 0 !!x = 0 !!

Your Assignment:Your Assignment:

PPaaggee 1 1552 2 16 – 42 even16 – 42 even

*43, *47 (show work for credit!)*43, *47 (show work for credit!)If you are a Pioneer, and you will miss Thursday, If you are a Pioneer, and you will miss Thursday,

then it is HOMEWORK!then it is HOMEWORK!

If you are NOT a Pioneer….If you are NOT a Pioneer….

Whatever you do not finish in class today Whatever you do not finish in class today can be worked on during class on can be worked on during class on

Thursday!Thursday!

TThhee EEnndd !!