8th Grade Algebra

this unit is all about solving algebraic

expressions now simple way to say what

an algebraic expression or equation is

is simply solving for an unknown

variable for example let's say I know

some number which I'm going to represent

by X plus 5 equals 15 now we don't need

to really do anything to take a guess at

what that X could be we know it's going

to be 10 so 10 plus 5 equals 15 so some

of these expressions are pretty easy to

figure out just by taking a guess at it

we're going to be looking at a few

harder ones where we got to follow a few

simple steps in order to solve for X so

whenever you have an equation it doesn't

matter what you have it could be x plus

5 all over 2 equals 15

whenever you have something like this we

ultimately want to end up with the

letter all by itself on one side and the

numbers all on the other side so if the

letter is e you can use s you can use M

it doesn't really matter what letter we

use sometimes you'll see X and sometimes

you'll see other letters so we always

want to figure out how are we going to

get this letter all by itself on one

side and all of the numbers on the other

side so we're going to do a few examples

with that and we're going to follow two

basic steps in order to do it so here's

one of the sheets that was included here

it's called infinite algebra 1 and what

we're going to try to do this is

starting off with a very simple equation

so I'm going to zoom in a little bit

just to make it a little bit easier to

see

oops this particular problem let me see

view zoom let's go three hundred percent

so it's a little easier there so here's

what we have we have 26 equals eight

plus V and in this case they didn't use

X they use the letter V

now we know we want to have V all by

himself in the end so what is it we want

to move

well this eight is going to have to move

to the other side so when we move him

over to the other side a funny thing

happens with the 8 since we know it's a

positive 8 because there's nothing in

front of it there's no a minus in front

of it when we move this positive 8 to

the other side we're going to still have

26 but we're going to end up with this

positive 8 becoming take away 8 and

don't forget we still have this positive

v left on this side and I'm just going

to leave it as V now when we go 26 take

away 8 the final answer will be 18 for V

so that will be our final answer

so in ones like this there's only one

step to do all you have to do is move

this guy over to the other side and

change the sign from a positive to a

negative looking at a 1 like this where

the P or the letter is on this side of

course we're not going to move the P

over here we're going to move the number

so this positive 3 goes over to this

side and becomes negative 3 so all we're

left with now is P on one side and of

course 8 take away the 3 on the other so

the answer to P will be 8 take away 3

which is 5 now we can easily test out to

see if it were right by simply putting 5

back into the original equation for P so

if I go back here and I look at the

original equation which was 3 plus P

equals 8 so remember our answer was 5 so

3 plus let's put 5 in for P and we can

put it in brackets if you want so 3 plus

5 does equal 8 so we have 8 on each side

so we know our answer is correct so

that's if you're asked to figure out

and of course remove this guy over

that's if you're asked to validate or

check your answer all you got to do is

take your answer in this case it's 5 and

plug it into your letter be careful when

you get to negatives here sometimes when

you put a number to the other side the

negative becomes positive of course so

in this case the minus 15 when we move

it to the other side to separate it from

the positive n we're left with n on one

side and this minus 15 don't forget the

minus 9 still there this minus 15

becomes positive 15 now the easiest way

to think of when you're doing addition

of negative numbers and positive number

you can take out your calculator in this

case and on most calculators you will

find a positive and negative sign there

so if I want a negative 9 all I have to

do is punch in 9 and then press the

positive negative 1 and that

automatically changes it to negative 9

and then to add 15 just go plus 15 and

our answer of course for this one would

be 6 so in this case n will equal 6 so

that's if you have a calculator that has

the plus and minus on it if you don't

just think about it this way if you have

negative 9 on a thermometer and I could

draw a lovely thermometer and if we're

all the way up to negative 9 of course

we go up a degree it's negative 8 if we

go up another degree it's negative 7 so

if we go up 15 degrees we're going to

have to jump 1 2 3

15 times we're going to do that 15 times

so you could draw a number line and

think of it like a thermometer and if

you're adding you go this way if you're

taking away you go the other way and of

course adding a negative if you have

one is the same thing as taken away so

in this case the 15 moves over here and

becomes negative 15 the plus 4 here

moves over and becomes negative for the

negative 9 moves over and becomes

positive 9 the negative 15 moves over

and becomes positive 15 and so on so in

just about every one whenever you have

it whenever you have it you're trying to

figure it out all you have to do is move

the one that's not attached to P

there's nothing attached to P just move

it over to the other side and change the

sign to a positive and you'll get your

answer here's another one it's a single

step one but it's not plus or minus

you'll notice that it's a fraction so in

any fractions written like that what

this fraction really means is divided by

and to find what it the opposite of

divided by is you just have to think of

multiply so if you have V over 8 and

you're dividing by 8 and you got it

times it by 8 or do the opposite that

will cancel it out for you but don't

forget whenever you do it to one side

you must do it to the other side so we

have V left all by himself

and 2 times 8 which is 16 so we just

solved for V V will equal 16 so I could

show you that one in another one here

there we go so we got 16 equals K over

11 I think I'll put that one on another

page so 16 equals K over 11 so we want

to get rid of what everything on this

side except for the K well since this

mean

divided by a fraction is divided by

we're going to do the opposite and we're

going to multiply by eleven that will

cancel out of course the 11s because 11

over 11 is going to equal 1 anytime you

have a fraction that is say even 7 over

7 that equals 120 over 20 equals 1 so

don't forget though when you times one

side by 11 you must do the exact same

thing to the other side so I'm going to

have the times 16 by 11 as well so I'm

going to take my calculator move him

over here and of course then I'm going

to go 11 times 16 and my answer of

course is 176 so my answer to this math

problem just by using the calculator

will be 176 left on one side and the

other side is just K so k equals 176 it

doesn't matter which side you put the K

on or which side you put the other one

on sometimes you'll see it written both

ways looking at this one this is not a

divided by there's no divided by

underneath that there's no fraction so

whenever you see just the numbers

touching the letter that really means

multiply and of course we're going to do

the opposite so when you got negative 15

times X we're going to go and divide it

by negative 15 like that that will

cancel out the negative 15 but don't

forget what you do to one side you must

do to the other so I'm going to divide 0

by negative 15 and it doesn't really

matter when we solve that zero divided

by anything is still going to be zero so

X will equal zero in this case looking

at number 19 there's a negative seven in

front of n so you've got to divide it

because negative 7 is times n so you

must divide it by negative

like that and of course you do it to one

side you can select the minus seven you

must do the exact same thing to the

other side so we're left with just n on

one side and twenty-one divided by seven

is three and you can use your calculator

for that but don't forget it's a

positive twenty-one divided by a

negative seven so a positive divided by

a negative is always a negative okay the

only way you can get a positive out of a

negative is if both of them were

negative which they're not so we're not

going to put it there so in this one

here since that's 14 times K we're going

to divide it by 14 on each side moving

over here this is negative 11x so we're

going to divide it by negative 11 in

order to cancel it out and then we're

going to divide this by negative 11 on

this side so a negative divided by

negative is a positive so we know our

answer is going to be positive so X is

all by himself now

143 divided by 11 if I were to take my

calculator so 143 divided by 11 and you

notice I didn't put in the negatives

when I did it because I know a negative

divided by a negative is a positive so I

don't need to worry about my negatives

what my answer does turn out to be 13 so

if we put X back in there if we put 13

times negative 11

we should get negative 143 so we could

always check our answer so this one here

is divided by 18 so we're going to times

it by 18 to cancel out this side of

course you must times this minus 5 by 18

to cancel out or sorry to get your

answer on that side so it's pretty much

the same thing all the time you're

either going to have the shift

if it's a take away or plus you're going

to have to shift it over to this side

and change the sign if it's a divided by

or a multiply you're going to have to do

the opposite end times it by seven or

times it by a number or in some cases

you'll have to divide it by a number

like we did up here all right so now

that we know how to do the shifting of

numbers and we also know how to multiply

the numbers sometimes you might get two

step equations and these are usually the

ones that are worth the most on a test

in these two steps you just got to do

one step at a time it's no big deal no

harder it's just you got to remember

which step to do first and in this case

we always move what's not connected to

the letter so this a over four we can't

just take the a off and move them over

here because he's part of a fraction so

we're going to keep him there and we

always moved the plus or the minus first

so I'm going to take this plus two and

shift him over to this side so I've

still got a over four left on the right

hand side on the other side since I've

moved my positive two over here the sign

changes to a negative two so six take

away 2 is four and we've still got the a

over four left now the easiest way to

get rid of a the divided by four that's

down here because that's the only thing

left that divided by four is two times

it by four to cancel it out and of

course if you times one side by

something you must do the exact same

thing to the other side so A's all by

himself on one side and of course

four times four will give you 16 on the

other so a must be 16

for this equation to be true looking

down here at this one

well maybe I'll do him on a separate

page again 9 X take away 7 so 9 X take

away 7 equals and just a double check 9

X take away 7 equals negative 7 so

remember the first step always is to

deal with the thing that's not connected

to the X first in this case it's the

minus or to take away 7 so I'm going to

shift him over to the other side and

he's going to become plus 7 because he

switches signs so we're left with 9x

all by himself equals negative 7 still

there and we've also brought over a plus

7 all of a sudden negative 7 plus a 7

they oh they cancel each other out to

give you 0 because if you have a

negative 7 degrees and you add 7 degrees

you're back to 0 degrees and of course

9x is still here and to get rid of this

9 times X we must d vide it by x and of

course we know 0 divided by anything is

still going to be 0 so X will be 0 in

this case and that makes sense because

if you put 0 back into X up here 0 times

9 is still 0 so that cancels it out and

we're left with just negative 7 on this

side which is what we've got on the

other side okay in this one here it

looks a little harder but it's not

really the first thing we deal with is

the plus or minus so we'll take the

minus 5 and bring him over here

so negative 5 now becomes a positive 5

because we shifted them so negative 4 is

still there plus 5 equals and remember

this is still on that side

our loops over 20 so negative 4 plus 5

if you punch that into your calculator

is positive 1

and that equals R over 20 the only way

to get rid of that divided by 20 now is

to do the opposite which is times up by

20 that will cancel it out and of course

1 times 20 is 20 so R will equal 20 now

I'm going to take this one V plus 9 over

3 equals 8 and I'm going to do that one

because that one's a little bit trickier

so V plus 9 over 3 equals 8 just the

double check make sure I wrote that down

correctly V plus 9 over 3 equals 8 so

again you would almost think you got to

move this plus 9 over here but the

problem is this V Plus 9 is part of this

whole fraction ok so it's like having it

in brackets okay so how do we get rid of

the divided by 3 first of all and the

easiest way to get rid of the divided by

3 is 2 times each side by 3 of course

that will cancel out the 3 and we'll

times this by 3 on this side then we're

left with V plus 9 on one side 8 times 3

if you work it out on your calculator

will give you 24 now it looks like one

of those one-step problems that we were

doing earlier so we take the plus 9

bring them over to the other side and

shift them and it becomes negative 9 so

V is all by himself 24 take away 9 will

leave us well you take away 10 in a

b14 so add one more and it'll be 15 for

our answer so remember anything in

brackets needs to stay together you

still times it by whatever you need to

times it by to get rid of the divided on

the bottom okay so let's try another one

and I think I'll try this one that

already has a 1 here so this one here is

two and then bracket n plus 5 now you

could go ahead and work it all out but

there's no really no need all you need

to think about is how do I get rid of

this two times all this stuff on that

side and the easiest way is to just

divide it by two and that'll cancel out

that but if you do it to this side of

the equation you must divide the other

side by two so we're left with n plus

five equals well two divided by two is

one and a negative divided by positive

is a negative so now all we have to do

is shift this plus five over to the

other side and it becomes negative five

that gets rid of it now we're left with

N equals negative one takeaway five more

is going to be negative six and that

will be our answer for that one

so in these two-step problems you always

start with the plus or minus first and

shift him over and then you've got it

times it by a number or sometimes you

got to divide it by a number depending

upon what the opposite is so as you can

see all of the problems on the example

sheets are pretty much the same you

shift them over and then of course

you're going to times it by four to

cancel eight the divided by four if

there's brackets well what's in front of

the bracket this is 12 or negative

twelve times all this so we're going to

go divided by negative

of 12 and that'll cancel that and of

course then it'll be much simpler to

solve be careful with your negative

signs

then you must shift this over first and

it becomes positive and then you've got

to divide by negative nine to get rid of

the negative nine times so don't forget

to do not just the nine times but the

negative that's in front of the nine as

well then the last one over here you got

to divide it by ten to get rid of the

ten in front but don't forget to do it

to both sides and watch your negatives

when you're doing this one and it should

work out for you nice and easy just like

the rest of them you might be given some

times the value of x and you might be

asked to plug it back into your equation

should be easy just take two and

wherever your letter is it doesn't have

to be X just simply put two or whatever

you're given into it and solve it like

you always do so 2 times 5 is 10 plus 6

more is 16 so the answer when you put X

in there will be 16 when you're looking

at this one here we could use why

usually we stick to X's and Y's in math

put 3 in for y and of course 4 times 3

is 12

take away 8 will give you 4 for your

answer and all the answers are given to

you anyway so this is just a practice

that you can go ahead and work on now

why do you need to know how to plug in

certain numbers and solve and get an

answer because when you start to

graphing coming up next you're going to

be given something with x and y and

you're going to have to plug in your own

numbers so

going to zoom back out just a little bit

and we'll go up to there and that should

give us enough so we can see it a little

bit clearer so here's what we've got

we've got Y and X so we've got two

letters now you remember back in grade

five and six you guys had to do these

coordinate planes it's almost like

playing a game of battleship where you

had to call up two numbers or a letter

in the number b-52 so you looked at B on

one side and you looked at 52 on the top

and then you match them up in order see

if you could sync some of the battleship

what you're really trying to do here is

come up with your own values and then

plug it into this chart going left to

right is called the x value okay so if

it's positive you go this way if it's

negative you go that way first so you

always go sideways first for X for the Y

value on this graph it's the line going

up and down so if your answer or Y will

end up being say 3 then you're going to

end up with your Y value of 3 so you're

going to end up with two things you're

going to have an x value separated by a

comma and a Y value so I could start

with X let's say X was 2 and I plug 2

into my formula here so if I put 2 in

for X 2 times 2 is 4 take away 5 is

going to leave me with negative 1 will

be my answer so I will plot this now

I'll start with my ex so you always

start from the middle and you say ok X

is positive 2 so I go over 2 and Y is

negative 1 so I go down 1 2 negative 1

and there's my first point plotted right

there ok but how do you know which

numbers to use generally whenever you're

given an equation and you're asked to

graph it the easiest way to do it is

just simply make your graph like this

put X on one side put Y on the other the

X is your input or the numbers you're

choosing and the y will be your answer

or your output so we'll start with

negative 2 negative 1 0 and then we'll

go with positive 1 I'll put the plus

sign there although you don't really

need it you know it's positive even if

it was just written 1

usually we pick a couple numbers below

zero and a couple above so let's start

with the first one let's plug in

negative 2 for X so our an equation

which is y equals 2x take away 5 I'm

going to plug in negative 2 because

that's my first one so let's do that y

equals 2 times negative 2 and then take

away 5 of course now let's keep going

and keep solving so Y still equals well

2 times 2 is 4 but since it's a positive

and a negative it's negative 4 take away

5 so it's going to become even more

negative so negative 4 take away 5

equals negative 9 so our answer here is

negative 9 for y so when x equals

negative 2 y equals negative 9 now let's

try putting X in as negative 1 so I'm

going to go down use the exact same

equation I'll rewrite it again down here

so remember our equation it is y equals

2x subtract 5 now of course I'll put in

my negative 1 so y equals 2 times

negative 1 take away 5 more so 2 times 1

is 2 since it's a negative it's going to

be negative 2 take away 5 more and my

answer for y

is negative seven so when X is minus 1 y

will be minus 7 I'm going to do one more

because 0 is really easy to plug into

any formula if we put 0 into here for X

it's going to be 2 times 0 well 2 times

nothing is still nothing take away 5

from nothing and we're going to be minus

5 so you notice the pattern that's

happening you can look at your first one

here is going up by 2 because it's

becoming a little bit more positive now

we're a little less negative whatever

way you want to look at it so in this

case it's going up by 2 and again it's

going up by 2 when you go from this one

to this one so really we could pretty

much fill in the next two numbers

without having to work it out so let's

add 2 more on to minus 5 and that'll be

minus 3 and then the next one when we

add 2 more it'll be minus 1 so it's a

little trick to solving the rest of them

I like to do 3 numbers because you can

see if you made an error in one of them

if you do 3 of them you can really see

and if you get the pattern the same you

know you're right so there's our

coordinates so let's start with X at

minus 2 so there's minus 2 and Y is

minus 9 of course I don't have enough

space here

I got minus 6 so I'd have to go down 3

more so that's minus six - seven - eight

- nine so I put my dot right there let's

do another one minus one minus seven so

we start there in the middle and go over

negative one for X and go down negative

7 for y which will be about right there

again your graph on a test will be

complete or we'll have enough now I'm

going to put in 0 and negative 5 so I'm

starting at X so I'm going to go over 0

well that's easy for X I just don't move

and then I go down 5 because I'm going

to negative 5

I'll plot another one it's going to be

positive 1 and negative 3 so I'm going

to go over positive 1 and down negative

3 and finally positive 2 so I'm going to

go over 2 and then I'm going to go down

1 and as you can see if I were to draw a

straight line providing I had done this

pretty accurately you notice all of it

should be in a straight line because

this is a formula for a a line so if

it's not all in a straight line you also

know that your answer must have been run

let's try one more here we have negative

6x plus 5 for this one you again draw

your table like this you put in your X

on one side and your Y on the other and

then you give your minus 2 minus 1 and 0

1 & 2 now do we have to pick those five

numbers it's not totally you could have

picked 1 2 3 4 5 but you do have to make

sure that all the numbers are in order

in this case they're going up by one so

if you had to pick one two three four

five it'll still work so the only thing

you have left to do is plug in minus two

into your equation right here for X and

then see what you get so if I put minus

two in here it's going to be negative 2

times negative 6 which is positive 12

plus 5 more is positive 17 now right

away you can see there's no way to graph

it on this you would need some graph

paper and you would have to make your

own coordinate grid because negative 2

and positive 17 would be way up here

somewhere

okay so that's not a very good numbers

to choose these ones right here the ones

you will be given of course will be much

easier to choose from and you just go

ahead and draw

your table with X&Y each time and then

put your numbers there and just plug

them in and once you get three of them

done you don't really have to do the

rest you can just see the pattern that's

going up here and keep jumping like that

so I hope this has been helpful for you

and if there are any questions or any

problems and you need some help with it

feel free to email me at mr. Edison at

hotmail.com well and of course nothing's

coming out on the computer so I'll put

it here so we'll be mr. Edison my

goodness shocking I'll try that one more

time okay so mr. Edison at hotmail.com

and that would be the easiest way to

drop me a line and if you have any

specific questions I can send you a

little tiny mini tutorial to help you

out