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How To Write Linear Equations
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Feb 19, 2025
In our last section( 2.3), we learned how to determine slopes/ or rate of change of lines. In Section 2.4, we'll learn first how to write the equation of a line given the slope and point of the line. We'll also learn how to write an equation of a line parallel or perpendicular to a given line. Chapters: 00:00 Introduction 00:39 Introducing the slope intercept form 01:56 How to write an equation of a line given its graph 06:45 How to write an equation of a line given its y intercept and a point 09:01 How to write an equation of a line given two points 12:22 Perpendicular and parallel lines' equations
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all right so we are going to start like
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writing linear equations and I'm sure
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you guys love the y = mx plus b a lot of
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people like that so this section should
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be fine okay now before we start we need
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to understand what what is a linear
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equation we've been talking about linear
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functions and linear equation for the
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last two weeks so a linear equation
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first of all it forms a line right
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that's why the word linear is there
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second a linear equation
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equation is always written in the form
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yals what mx + b that should be like
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second nature now we should know this by
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Heart Right algebra one prealgebra we've
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learned it so a linear equation is
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always in the form yal MX plus B and M
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and and m stands for what the slope
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right the slope is it tells you what
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direction the line is going is either
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going up down uh uh slanted is it flat
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so the slope gives you that right it
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gives you the direction of the line and
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then B is the Y intercept the Y
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intercept meaning where the line crosses
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the Y AIS right so these are the most
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important one m and b they always going
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to be there
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now B the Y intercept is written this
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way zero and B Because when you have a y
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intercept that means that the line is
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crossing the y axis and at that specific
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point the x coordinate is what zero this
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this is why he crossing the y axis CU
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you you're going through the y axis that
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means the x coordinate is zero right now
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this equation here y = mx + b is called
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the
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slope intercept from slope incept form
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that's what you call it all right this
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is what you call that y = mx plus v now
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the first thing that we're going to do
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with this equation is we're going to
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learn how to write an equation given the
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graph of the line so when you see the
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graph of a line can you graph an
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equation given the graph of a line right
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we're going to learn how to do that
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because we know now that every line is
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in the form what y = mx + b correct so
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now I have this line here right before
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we even start if you look at the Blue
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Line right do you think the slope is
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going to be positive or
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negative negative why it's going it's
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going down right so we already know it's
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going to be a negative slope that's the
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first indication right so that means if
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you find your slope and it's positive
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that means that you did something wrong
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right so always start with that you know
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that the slope is negative now that we
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know that the slope is negative the best
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way to do this is this we need to find
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the Y intercept is the easiest to find
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isn't it what's the Y
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intercept uh
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Co what's the Y intercept on this line
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your blue line what's the y intercept
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um
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negative what does that mean the Y
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intercept like where it starts kind no
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what's the Y who knows the definition
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yeah um it's -2 cross I
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said2 no you Saida one or something I no
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I said2 so did you mean this side this
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here yeah okay but if that's yeah all
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right so where the line crosses the Y
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AIS wait I
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right I right you got it right why why
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did you say no I don't know I I don't
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think I heard -2 so where the line
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crosses the y axis right right here 0 so
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0 -2 okay now what is the letter for the
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Y intercept in our equation K what's the
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letter for the Y intercept in the
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equation x that's not x what is
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it b b we talk we just talked about B is
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the Y intercept in the equation B is the
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Y intercept right so that means B is
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equal to
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what -2 -2 so now we know that b is
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equal to -2 so we know that right so we
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can build our equation now we're going
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to go y = mx + -2 because we know the Y
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intercept right now what is missing so
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that our equation is
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complete what is missing slope the slope
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now how do I find the slope you
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do
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uh 1 + 2
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over uh2 right 0 - 2 yeah because to
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find a slope you need to have any points
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to find the slope two points right so
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since we already have this I've already
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given you this one we have 0 -2 and -2 1
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now we can find the
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slope because now we have everything
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right so we're going to go the slope
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formula this is going to follow you for
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a long time until you leave the school
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and probably go to college you're always
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going to talk about the slope right m is
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Y 2 - y1 over X2 - X1 this is how you
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find the slope right now my Y2 in this
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case is I chose -2 1 X2 Y2 and y1 X1 I
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chose -2 and zero does that make sense
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are we good right and now I can find my
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slope so I'm going to do -1 - -2 now be
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careful because on the homework
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assignment that I just gave back some of
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you guys have the X up and the y in the
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denominator you have to really be
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careful when you do that you need to be
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able to recognize what's Y 2 what's y1
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because if you don't do that you're
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going to get it wrong and then once you
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you the slope is wrong everything else
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is wrong so you got to make sure you do
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that correctly okay so our slope is -2
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so the rest is just plugging the slope
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value right so now that we know the
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slope is2 we're going to replace it or
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substitute it in the equation and now we
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have y = -1x + -2 or simply -1x - 2 does
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that make sense okay so again recap find
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this find the Y intercept if it is given
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in the equation and then find the slope
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all right now case number two how do you
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write the equation given the slope and
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the point
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here we did what we had the Y intercept
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and then the point right now we have the
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slope and then we have a point so how do
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you find the
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equation right so we start again our
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starting point is always going to be
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what y = mx + b always start with your
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foundation right at the outset you
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already know that this is a linear
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equation so this is going to be where we
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going to start does that make sense
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yes or no yeah all right so if I know y
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= mx plus b now what do I go from here
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sh what I know that this is my equation
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Y = MX plus b I know slope is4 so what
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do I do first you plug it in I plug it
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in right yeah I have 4x + B now what am
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I missing your b b so how do I find
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B is the question right how do we find B
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we know that this is -4x + B now here's
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what we do we know that the line is
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going to go through what 6
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and -2 right so to find B we're going to
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replace X by what 6 and Y by -2 and
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solve
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for b right easy stuff so you going to
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go -2 = -4 * 6 + B why are we doing this
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because we know that the line is going
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through what 6 and -2 so we can choose
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to plug here to solve for
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b does that make sense and now when we
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solve it so we know -2 is -4 + b b just
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going to be plus 24 B is 22 and then you
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replace it back into your equation right
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once you find B now we know what B is in
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first we know B we can replace it and
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substitute it and B is POS 22 so the
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equation is -4x +
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22 does that make sense all right so now
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case number three we're going to have
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another case is when we have two points
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and now how do we graph the line okay so
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let's say I have -27 can I raise this
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one in the middle yeah all
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right so basically how to write an
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equation
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given two points
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right and
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let's assume that we have the two points
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are uh
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-27 and then three and -3 right so I
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have two points like this and now I want
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to graph my equation where do I start
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what's my Foundation
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again pck where do I
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start so I I first have to write what
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down my equation right what is it y YX
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plus b you start here always right now
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what am I missing in this
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case huh both of them right I'm missing
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what M and I'm also missing B right I'm
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missing both of them so the first thing
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I can find is what I can find I'm giving
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two point what can I do here I can find
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not graphic slope slope right I can find
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the slope and how am I going find
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it uh tell me how am I going to find
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it y all
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right all right so we going to call this
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Y2 X2 right so we're going to go 7 - -3
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right over what -2 - 3 so that is 7 + 3
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that's 10 over5 so the slope
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is -2 right the slope is so we know we
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go back -2x + B now we are only missing
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one piece which is the B right so to
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find B what are we going to do we're
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going to do use the same method that we
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use here I can choose either one of
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these it doesn't matter which one I
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choose right I can choose this or I can
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choose this point because I know that
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the line is going through e both of
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those points anyway so let me say I
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choose 3 and3 so I'm going to replace X
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here by what
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what three and then y
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by3 right so I'm going to go -3 = -2 * 3
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+ B and then solve for b right so that
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is -3 = -6 + B and we going to add six 6
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so 3 is equal to B so now I have B now I
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can write my equation because B is known
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so Y is =
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-2X do you need to do the one or can you
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do the two I can do either one it's
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going to give me the same answer you can
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choose this or that because the line is
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going through either one of those so the
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outcome is always going to be the same
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all right so the outcome is is going to
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be the same because it doesn't matter
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which one I choose I'm going to get the
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same thing all right now we're going to
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talk about
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two other things right we're going to
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talk about can I erase this
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we're going to talk about perpendicular
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lines and parallel lines okay
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perpendicular and
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parallel
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perpendicular and then par
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lines so what do you know about
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perpendicular lines what do they have if
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two lines are perpendicular what what am
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I saying um yeah they intersect of the
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right angle they intersect at the right
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angle right so perpendicular line will
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look like this they're going to
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intersect at a right angle right that's
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perpendicular line okay what about
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parallel
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lines somebody else beside JD they're
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beside other never touch they never
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touch right they never touch those are
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going to be parallel lines they're going
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to be like this now there's a
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relationship between the two
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right with line they have what the same
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what line have the same SL slope so if I
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have y = 3x - 1 and y = 3x + 2 these two
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lines are going to be parallel lines
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because they have the same slope so in
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other words parallel
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line lines
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have the
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same slope that's just a given it's
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always going to be true no matter where
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you go China Africa wherever you go par
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lines are always going to have the same
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slope it doesn't matter where you go
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this is an objective truth right they
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always going to have the same slope what
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about perpendicular line what's the
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relationship between their slope yes sir
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does that what you do with reciprocal
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reciprocal and what else 0 no reciprocal
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and negative opposite reciprocal and
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opposite yeah so let's say for examp
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have y
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= 14 x + five the if a line is
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perpendicular to this line the slope
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will be
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-4x over 1 + 5 right that inverse and
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opposite be yeah but I just did it to
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just show what's going on I could just
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leave at4 but I wanted to show that you
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actually reverse it right so for example
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if I have a line with the slope n = 23
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and I want to find a line that's
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perpendicular to this line what would be
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the slope of that new line uh3 to 32
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right 32 because you reverse it and it's
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negative or opposite right cuz if this
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is opposite then will be positive that
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one will be positive but reversed does
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that make sense yeah now we're going to
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try to build an equation of a line using
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this information so let's assume
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that um
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time is class uh I don't know
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25 really 2 oh we just warming up we
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just
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warming we just warming up end actually
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just goes right to the beginning like
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Groundhog Day
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oh we
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just we're just warming up right all
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right I hate that
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movie can I get water goe so what what's
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the what's the
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purpose what's the purpose of a
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groundhog I don't know what's the
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groundhog doing I know he comes out of
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the hole and tells us that we're going
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to have a early winter or
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whatever he's trying his best he's
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accurate anyway hold let me finish this
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so we can go back to let me finish this
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real quick all right so here's the thing
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now let's say I want to find a line
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right that's perpendicular this line
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find the
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equation find the equation Mr yeah why
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are you all dressed up today I'm not
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dressed
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up Ain dressed up I ain't got no dress
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shoes
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on that's
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prepic to this to the line
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above and that
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goes through
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5 -6 so we want to find a line that's
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perpendicular to this line right and
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that goes to 5 -6 so where do I start
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what's my starting point
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Nate what's my starting
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point
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M before we do that before we even do
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that yx+ b right I I always want to
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start here that's point1 now i' I've
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been given some information I want to
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find the equation of the line that is
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perpendicular to this
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line right so what does that say about
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the slope of that line it's going to
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be it's going to be 2/3 there you go 2/3
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that's perfect so the slope M will be 23
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because it's what it's inverse and
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opposite right so be y = 23 X plus b now
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I'm only missing what I'm only missing
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one thing here what am I missing the Y
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intercept y intercept I find it replace
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5 and6 that's it we replace X by 5 and Y
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by -6 right so we're going to go 23 * -5
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+ B okay now we have this so we have
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-6 oh yeah posit 5 you right so that
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would be 10 over 3 + B right now check
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this out so I want to get rid of the
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fraction right cuz I don't want to deal
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with a fraction so what do I do here
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multiply by multiply everything by three
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right from last chapter everything by
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3 so we get -8 = 10 + 3 B and then we
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just solve for b from here does that
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make sense the
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-10 -10 so I get
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-28 = to 3B and then divide this by
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3 so B is 28 3 so Y is =
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23 x - 28 / 3 and that's the equation of
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the line does that make sense right and
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if you wanted to find a line that is
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parallel to this line going through this
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you would do the same thing but the
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differences the slope will be what the
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same right when they parallel line the
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slope is the samees that make sense
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parallel lines same slope perpendicular
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line reverse and opposite all right so
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that concludes this section so remember
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this week we're going to have a
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