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How To Solve systems Of Equations Graphically & Algebraically
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Feb 19, 2025
In our last chapter, we learned how to graph and solve linear equations. In chapter 3.1 we'll first learn how to solve systems of linear equations graphically but also algebraically. Chapters: 00:00 Introduction 02:14 Solving Equations By Graphing 05:50 How to classify systems of equations 10:10 Solving a system of equations with the substitution method 14:10 solving a system of equation with the elimination method
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0:00
welcome back to Algebra 2 we are on
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chapter
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3.1 and we're going to talk about
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solving system of equations all right
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now before we even get to that the first
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question we must ask is this what's an
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equation an equation by definition is a
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mathematical
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sentence uh that states that two
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expressions are equal right so when you
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have a sentence that states that two uh
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expressions are equal you have an
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equation now more importantly in this
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case we have two we have a system of
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equations we can have two equations
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three equations four equations and when
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you have those type of equations we
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usually use matrices but for this kind
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where we only have like two equations
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we're not going to use a matrix to a
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matric to solve this all right now by
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definition a system of equation with
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this uh a system of equation is a system
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of equation with the same variables
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right and why do we use this can we use
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this in real life absolutely yes and I'm
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going to give you an example suppose
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that you're trying to you're trying to
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run a business right maybe you you want
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to run a business you want to chop down
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trees and you want to get a machine
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that's going to help you to do that
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right you want to turn them into like w
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chip that they put on playgrounds and
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stuff like that right so let's say you
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invest like $400 in the
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machine uh that's just the initial price
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and then every month they require to pay
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a certain fee for maintenance right and
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now you want to see uh to to to chop
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down it you charge people a certain
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amount of money so you want to know how
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much do I need to charge people to break
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even right so you're going to be using
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equations to do that so you're going to
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use them and it's very helpful all right
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I do that in my own field when I told
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you about when I write articles and
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things like like that so now the first
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thing we're going to learn is this how
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do you solve this equation by Gra
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right you have a system of equation and
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the system of equation here we usually
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put this to show that there's two
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equation that we are trying to solve at
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once right now when you look here I have
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2x - y = -1 and I also have 2 y + 5x is
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equal to -16 right and I want to solve
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this so to solve this we need to find
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what we call an ordered pair right to
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solve the system we must find the order
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pair that satisfies all the equations
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you want to find an ordered pair in X
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and Y that when you plug it in here and
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plug it in here you get the same result
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that mean it satisfies both equations
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right so now to solve this there's two
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three ways to do it right we can solve
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by graphing we can solve by using the
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substitution method we can also solve it
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by using the uh elimination method now
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the first one we're going to learn is
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we're going to learn how to solve by
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graphing right so we're going to graph
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each of these graphs each of this
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equation and we're going to find where
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the intersect and the intersection point
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is going to be our solution right so to
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to to graph this function you just have
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to solve for y I have to solve for y
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right I have to put in terms of Y so I
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can solve it so if I have 2x - y = -1
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first I need to subtract what 2x -2X
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yeah yeah 2x right so we get- Y =
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-2x - 1 and because everything is
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negative I have to divide by negative 1
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so I can get a positive value here right
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I get 2x + one right that's my first
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equation the second equation I have 2 y
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+ 5x = -16 right I'm going to solve for
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y I'm going to do the same thing so I'm
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going to first subtract 5x so the first
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one is done second one 2 Y = 5x - 16 and
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to find y I'm just going to divide it by
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2 here right so I got y = 5 2x - 8 so
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this is it here right so now we have two
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equations so we're going to graph them
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and then we're going to find where the
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intersect the intersection is going to
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be the solution right so let's let's
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graph them so the first one the Y
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intercept is one and the slope is 2 over
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1 2 is 2 over one right I'm going to go
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up to one two and over to the right one
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right so that's my first equation
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and now I need to graph the second one
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the second one is the Y intercept is8 1
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2 3 4 5 6 7 8 and then is5 over 2 so I'm
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going to go down five unit from here 1 2
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3 4 5 and over to the right two unit one
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two right here
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right and I'm going to draw this see
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pretty much put it into y plus and then
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graph it graph it and now to find the
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solution solution is going to be here my
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point of intersection so this looks like
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1 2 3 and then 3 three so my solution
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is33 -3 right -33 is my solution because
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that's where the intersect to double
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check that you can plug this in here and
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then plug this in here it should give
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you1 and then if you plug this here and
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here it should give you -6 now my graph
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is not quite accurate because I didn't
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use like accurate things so you may be
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wrong but you you get the you get the
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the point the gist you trying to find
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the solution right to find where it
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intersect so this is how you solve an
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equation by graphing you just find the
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intersection point right now there's
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another thing that we need to learn how
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do you classify a system of equation
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right so to classify a system of
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equation we have four main components
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here we have consistent we have
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inconsistent we have independent and we
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have dependent what does this mean the
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best way to learn is to have a visual
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right with a visual it's easier to
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interpret so now when is a system
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consistent a system consistent is
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consistent when it has at least one
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solution right it has at least one
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solution now if you look here does he
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have at least one solution yeah yes so
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it's consistent right but that's not
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where we stop we can go a little further
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right when the system has exactly one
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solution we say that not only consistent
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but is also what independent right so
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this system here this first equation
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that we solve is
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consistent but it's also in dependent so
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it qualifies for both of this world it's
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consistent and independent because it's
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consistent because we have at least one
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solution and it's independent because we
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have exactly one solution would an
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infinite number of solutions just be the
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exact same equation yeah
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when they lie on top one another so that
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means it's it's infinite number of
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solution because the intersect for an
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eternity right so for example but let
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before we get to that what if I have an
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equation like this
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right let me draw something here y = 4x
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+ 1 and y = 4x - one right let's graph
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this here so 4X so when X is one here
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I'm going to go up 1 2 3 4 and then over
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to the right one
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right and then let's do another one here
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4x - 1 I'm going to start from here
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negative 1 right 1 2 3 4 and then
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one uh it wrong I think yeah I did
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something wrong
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here1
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and 1 2 3 4 it's supposed to be two par
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line so I didn't quite accurate it's not
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quite accurate you see they're going to
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be parallel lines right because they
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parallel line we can say what the system
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is what in inconsistent why because they
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have what No Sol no Solutions right
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there's no way they intersect they don't
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intersect anywhere so the system is
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inconsistent because they don't
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intersect there's no point of
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intersection and then therefore they is
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inconsistent yes sir um would it be
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possible to have more than one solution
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and it not be dependent yeah is that the
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curves it would be one for
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example absolute value yeah well if you
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have something right if you have
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something like that we we not going to
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discuss this in this chapter because I
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don't think we do a lot of these like if
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you have a function like this right and
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then we have a function that there's
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maybe this goes down here and
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this you see this now you have how many
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solutions two because you have it's
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consistent and it's also it's consistent
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won't be independent what what would the
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equation of the curve yeah it could be y
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= uh square root of something but we not
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doing that in this chapter just yet all
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right
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yeah all right but now now let's look at
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this when you have when you have two
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lines right if you draw two lines and
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then one is like this and the other one
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is also
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go the same directions right basically
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this is two lines in one right they're
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the same line so that means that the
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system is what is dependent because he
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has an infinite number of solution
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because the line the the lines are like
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that they're intersecting multiple times
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for all the way to Eternity so that
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you'll be it's consistent and is
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dependent consistent because he has at
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least one solution and and dependent
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because he has an infinite number number
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or solution so this is how you classify
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them right now the next thing we're
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going to do is we're going to move on
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and talk about uh using the substitution
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method to solve an equation substitution
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so that one is fun all right we going
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talk about the substitution
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method this is fun
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too
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so the substitution method substitution
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right substitution mean you substitute
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something into something else to solve
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for something
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else so let's say we have 5x right - 3 y
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= to
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23 and then we have 2X + y = 7 and the
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question is asking us to use the
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substitution method to solve the system
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of equation so here's what you do right
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you can isolate one of the variables and
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then use what you have to plug it into
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the another one right so here for
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example I have 5x I have 3 y I probably
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won't use this as my help I'm going to
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use this one down here because here I
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can solve for y easily it only take me
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one step right y will be equal to what
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zero who say zero I don't know 7 - 2 7 -
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2x right now we know what Y is right so
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now I'm going to choose this and I'm
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going to replace Y in this equation by 7
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- 2x right this is called the
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substitution oh yeah we've done this
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before you've done it I'm sure you did
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so now we have 5x - 3 instead of Y this
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is not going to be 7 - 2x = 23 but where
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is this like used in math like where is
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it used like where like is this used for
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graphing graphing is also used for
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finding Break Even point Point things
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are the sword all right so now we can
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solve this so it's 5x now - 21 + 6 x =
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23 and then we solve for x 5 x + 6 x
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that's 11 x - 21 = 23 right then we add
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21 add 21 we get 11 x is = to
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44 4 and then X is 4 right X is 4 now
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that we find X we can solve for y can't
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we
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yes because we know that Y is what 7 -
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2x so I'm just going to choose this and
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plug it in here I go back here say Y is
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7 - 2 * 4 right Y is therefore 7 - 8 y
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is1 so now you have X you have y you
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need to write your solution as an
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ordered pair right it's going to be four
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and
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what1 like this like an order pair right
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you write it like an order pair at all
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time this is how you write this so now
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would you say that this is consistent
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yes or no is it consistent is it
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dependent oh wait what do we say uh it's
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consistent but it's not dependent it's
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not depend he's independent because he
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is consistent he has at least one
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solution and because he exactly one
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solution is independent all right so
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they can ask you that so this is the
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substitution method substitution okay
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substitution
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and now the next one we're going to
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learn is called
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the uh elimination method elimination
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okay do we have to use all these methods
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well it depends on well you can pick a
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favorite when the question is out in the
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open like solve the system but if I say
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solve by graphing you got to do by
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graphing if I say solve by using the
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substitution then you have to use the
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substitution if I say solve by using the
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elimination then that's what you do now
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only have three I hate elimination
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elimination is actually fun right
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now let me show you a technique here
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that you can use to to do the actually
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can we use this can we use the ignition
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to do the same problem let's try that
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right yeah let's try that so you see if
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you like this better so let's say we
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have the same problem and now we using
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the uh elimination method right
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elimination all right so I have
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5x - 3 y = 23 and I have 2X + y = 7 so
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the way you do it is this you look at
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your variables right which one is here
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is easier to get rid of the y or the
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X the Y the Y right so I have -3 y here
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in order for me to get rid of the Y this
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has to also be equal to what three so
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that means I'm going to multiply this
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equation by three right I'm going to
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multiply this equation by three so now
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now I have 5x - 3 Y is 23 and down here
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I have 6X + 3 y = 21 what the problem is
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is a lot of people when they multiply
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they forget to multiply the entire
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equation make sure you don't do that
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right so we multiply the entire equation
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by three now we can go ahead and just
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proceed by elimination right so now if I
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draw my line
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here3 Y and 3 y That's gone isn't it
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over right and I have 11 x here = 44 and
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to find X just divide by 11 x is equal
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to 4 right and then once you find X you
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can use this and plug it in here to
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solve for y what so what happens if the
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Y is nothing X the Y is nothing what you
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mean like like if if the Y at the top
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was just y why not negative like just
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one you could just multiply this by
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negative one the goal is to always get
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rid of one of them right and then once
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you find y x you can plug it in here now
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let's do this again let's do the same
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problem right 5 yeah does it matter what
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equation you put four no doesn't you can
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put in either one of those you get the
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same say say I didn't want to get rid of
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the Y but I wanted to get rid of the X's
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right so what do I
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do the opposite of one right so since I
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have five and two here I'm going to
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multiply this one by what by -2 and and
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this one by five you see because to get
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the 5x I find basically I'm finding the
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what the LCD kind of like this is kind
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of like what you're doing you're trying
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to find a number that it can multiply so
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that it can they can be opposite of one
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another right if I multiply 5 by -2 I
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get -10 If I multiply 2 by 5 I get 10
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and now they cancel out and I can use
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the same process yeah yeah so if I
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multiply this let's do it we get 10 x
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right + 6 y = -46 and here I get 10 x +
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5 y = to 35 right now watch the 10x are
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going to cancel out we're going to get
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11 y here equals to -46 + 35 that's -11
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wait is this probably the same yeah I'm
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just I'm just considering using the Y
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now and then / 11 Y is1 and if you plug
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this back in here you're going to find
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that X is four right so this is the
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elimination method now this is the
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pretty basic ones so you going to have
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different ringles and then you can use
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that and you going to have a word
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problem you can use that if I specify
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using the substitution the user
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substitution if I say use the you use
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the graph then use the graph and if if I
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say use the elimination then you use
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that right so basically these are the
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three that we going to learn in this
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chapter so now I'm going to assign some
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work that you can doing class now we got
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about 25 minutes just to if we don't H
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if you finish it if you don't
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