When we are trying to find the graph of a linear equation here�s one thing we can
use to our advantage, we understand that any point on the graph of an equation is gonna
be in a ordered pair x, y that has the special property that this x and this y when you plug
it in are gonna make the equation be true now points that are not on the line like here
and here these x, y ordered pair would not make the equation be true so when you look
at a line like negative x y equals negative x plus 4 you will have points on it like 0,
4 because if you plug in zero for x and 4 for y zero plus 4 really does equal 4 and
one three and two two and three one and four zero and the special property that all these
points have if you plug in the x and the y simultaneously into the equation they are
gonna make the equation to be true and so we can use that to our advantage to find the
graph of linear equations like this so here is some steps that we can do to find the graph
of a linear equation if we want to do it using points so step number one is first of all
solve the equation for y if it�s not already solved for y a lot of these equations like
this one will already have y equals and if it doesn�t have y equals then we should
make it be y equals using algebra adding and subtracting, multiplying and dividing, next
we are gonna draw an x, y table what we are gonna attempt to do is find a list of ordered
pairs that make the equation be true like this list of x, y values and so right now
we are not gonna find the points right yet we�re just gonna draw the empty table so
just draw a vertical line with 2 columns an x column and a y column now to actually fill
out that column here�s how we gonna do that we you and I are gonna choose some x values
these are completely of our own choice there�s no right or wrong x value you have to pick,
just choose any x values and then what you will find is that there are some y values
that will complement those x values that will make the equation be true let�s look back
at our previous example for instance let�s look at let�s look at maybe this point right
here so if we just simply chose to let the x be 3 and we did not choose a y value to
go with it look at what would happen we would have negative 3 plus 4 and then this will
tell us hey we really need to let our y values to equal 1 if we want this to be a solution
so what happens is we will choose the x�s and then we will let the algebra figure out
what the y values need to be that will complement those x values and what we will get after
we plug in those x values to find the corresponding y is we will get a whole table full of x y
ordered pairs a whole table of points and so the last two steps are to plot the points
take those order pairs and put them on the graph now what should happen when we plot
these ordered pairs well again looking back at this picture we have here if done correctly
they should line up in a straight line and so that should very clearly reveal what the
graph of the linear equation should be its simply the line that goes through those points
that you found so the last step we have number 6 is to simply connect the dots and you got
your graph of your linear equation now this method here this is only one method one way
of graphing a linear equation, there are a couple of other ways we can graph linear equations
that we will look at in the next videos this particular way has a few benefits a few pros
and a few cons, so let me highlight just the big ones, the pros of graphing lines this
way is that it�s very straight forward and easy you pick some x�s and you find the
y�s that go with them and you plot them and it shows you your line its very very low
very low amount of errors that are typically run into with this type of technique because
it�s even self-correcting believe it or not let me go back to this picture here let�s
say you were making up an x,y table and you found the first point ok and you found the
second point ok let�s say this one was fine and this one was fine and this one was fine
but let�s say we made an algebra mistake and messed up on the third point and maybe
put our dot out here for some reason, we accidently did a minus instead of a plus or something
like that well you see very quickly that these don�t line up anymore, so you say oh wait
I must have done something wrong with this ordered pair you go check your algebra you
see your mistake its fixed then you put it back where it should be and then you�re
off on your way so it�s very very difficult to make too many errors doing this method
unless you know unless you make some consistent algebra mistake over and over again repeatedly
but usually you will spot your mistake if you only make one mistake or two mistakes
or something. The down side, there is a negative to this method is that this is very time consuming
u know doing 6 steps to graph a linear equation is kinda a lot because you have to solve for
y make your table pick the x�s plug them all in find the y�s plot the points connect
the dots it�s just very time consuming and it turns out some of the other techniques
that we are gonna learn coming up are a lot faster and so you know if we were to come
back and graph some linear equations toward the end of the course this would probably
not be the ideal way of graphing lines, but as we are just learning how to graph linear
equations this a great way to begin to learn how to graph linear equations so let�s wrap
up this video with and example let�s say we are trying to graph the linear equation
y plus 2x equals negative 1 so let�s run through the steps real quick step 1 is to
solve for y so we take this guy and write it as y equals we will have to subtract the
2x to the right y equals negative 2x minus 1 so this will be the equation we�re gonna
use for the x, y table. Alright, next up we are gonna make that x, y table so we will
so a vertical line to separate out 2 columns like so and then x column and a y column and
then we just need to pick any amount of x�s and it doesn�t even matter what the x�s
are, just to be consistent. Most people will go from minus 2 to 2 its absolutely not necessary
but it�s nice to pick points around zer0 so that your graph is centered around the
origin. Technically we could have picked the x�s like 11,12,13,14,15 but then our graph
would have been way over here. And normally when you draw your x, y axis you are gonna
see the origin right there in the center, so now these are the x�s but I don�t know
if the y�s that go with those x�s so the next step would be to take these x values
and plug them in for x into the equation so now it�s actually clear why we solve for
y before we begin this way if you have y solved for when you plug in for x you immediately
get your answer you immediately get your y if you plugged In x into the first equation
every single step you would have to solve for y repeatedly but this makes it a lot easier
so let�s run through algebra negative 2 times negative 2 is 4 ,4 minus 1 makes y equals
3 so we will have negative 2 comma 3 that would be one ordered pair on our graph so
let�s lets so ahead and starting making our graph as we find these points so well
plot negative 2, 1,2,3 so that point will be plotted right here alright next up plug
in negative one negative 2 times negative one is positive two ,2 minus one makes one
so we will plot negative 1, 1 that�s right here next up plug in 0, zero minus 1 is negative
one so we will plot 0 comma, negative one which is right here then plug in one negative
2 times one is negative 2 negative 2 minus one makes negative 3 so we will plot one comma
negative 3 and last point plug in the next value of 2 negative times 2 makes negative
4 minus one make negative 5 so 2 comma negative 5 is down here and sure enough if you do it
correctly do you see how they all line up in a row, if we had made a mistake on one
of these points let�s say we messed up on our algebra we plotted a point over here then
notice these would not line up in a straight line anymore so we would go fix our algebra
but everything looks fine here so the last step is to draw the graph of this linear equation,
all we simply do is connect these dots that are already in a line so that blue line represents
this linear equation and now we don�t even need these highlighted dots now just the last
point I want to make here about this graph here you notice there are other points on
this graph other than the points we have highlighted like here and here and here and you know all
these places here those are additional x y point that are not in our table but they still
would have made the equation be true so these x y ordered pairs are a great way to begin
to learn how to graph linear equations.