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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
0:35
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
1:01
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
1:15
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
1:45
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
12:51
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
14:04
can we use this can we use the ignition
14:06
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
14:53
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
15:11
they forget to multiply the entire
15:13
equation make sure you don't do that
15:15
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
15:29
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
15:46
Y is nothing X the Y is nothing what you
15:50
mean like like if if the Y at the top
15:52
was just y why not negative like just
15:56
one you could just multiply this by
15:58
negative one the goal is to always get
16:00
rid of one of them right and then once
16:02
you find y x you can plug it in here now
16:04
let's do this again let's do the same
16:06
problem right 5 yeah does it matter what
16:09
equation you put four no doesn't you can
16:13
put in either one of those you get the
16:15
same say say I didn't want to get rid of
16:17
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
16:29
this one by five you see because to get
16:33
the 5x I find basically I'm finding the
16:36
what the LCD kind of like this is kind
16:39
of like what you're doing you're trying
16:40
to find a number that it can multiply so
16:44
that it can they can be opposite of one
16:46
another right if I multiply 5 by -2 I
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get -10 If I multiply 2 by 5 I get 10
16:52
and now they cancel out and I can use
16:54
the same process yeah yeah so if I
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multiply this let's do it we get 10 x
16:59
right + 6 y = -46 and here I get 10 x +
17:06
5 y = to 35 right now watch the 10x are
17:11
going to cancel out we're going to get
17:12
11 y here equals to -46 + 35 that's -11
17:18
wait is this probably the same yeah I'm
17:20
just I'm just considering using the Y
17:22
now and then / 11 Y is1 and if you plug
17:27
this back in here you're going to find
17:28
that X is four right so this is the
17:31
elimination method now this is the
17:34
pretty basic ones so you going to have
17:36
different ringles and then you can use
17:38
that and you going to have a word
17:39
problem you can use that if I specify
17:42
using the substitution the user
17:43
substitution if I say use the you use
17:47
the graph then use the graph and if if I
17:49
say use the elimination then you use
17:52
that right so basically these are the
17:54
three that we going to learn in this
17:56
chapter so now I'm going to assign some
17:58
work that you can doing class now we got
18:00
about 25 minutes just to if we don't H
18:03
if you finish it if you don't
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