here are the top 10 most important things to know about linear relationships must know number one what do linear relationships look like let me show you how you could recognize a linear relationship from the three ways that any relationship can be represented with an equation with a table and with a graph an example of an equation of a linear relationship would be y = 2x + 1 any linear relationship is just a variation of this one where the dependent variable Y is equal to a constant multiple of the independent variable X and that constant being multiplied by X we often call the slope coefficient plus some constant added to the X also notice that the exponent on the x is a one which makes this a degree one polinomial function if I were to make a table of values for this equation I could choose any values I want for x I'm going to make sure I choose values that are evenly spaced so I'll choose from -2 to two going up by ones and then I'll calculate their corresponding values of Y by taking each of those X values subbing them into my equation that tells me the relationship between X and Y so for example if x is -2 y would be 2 * -2 + 1 which is -3 I could do that for each of the X values I chose and I would get each of these corresponding values of Y from this table I should be able to tell that I have a linear relationship and I can do that by analyzing the difference in consecutive yv values so if I look at the first pair of Y values and find the difference by doing the bottom one minus the top one Nega 1 - -3 that's two the difference in each pair of consecutive y values notice it's two each time so that tells me as the X values are increasing by one the Y values are increasing by two because they're increasing by a constant amount I know it's a linear relationship all linear relationships have constant first differences and if I were to graph each of the points that I have in my table of values here notice that if I were to connect these points they form a straight line linear relationships are called linear relationships because they form straight lines when graphed so now you know how you can easily recognize a linear relationship from either its equation its table of values or its graph must know number two slope you have to know how to be able to find the slope of a line or another word for slope is the rate of change we're going to look at the graph and table of values for the linear relationship Y = -23 X + 1 Let's Start by finding the rate of change or slope of this relationship from its graph so if I look at its graph when I say I want to find its rate of change I really mean I want to find the slope of the line and I use the letter M to represent slope and to calculate the slope we have to figure out what is the quotient of the change in the Y values divided by the change in the X values so as X is changing what's happening to Y to calculate that we have to pick two points on our graph any two points will do but make sure you pick points that you for sure know the coordinates of and then what we want to analyze is how are the Y values changing between those two points which means what is the vertical distance between those two points and we actually have a word for that when we're talking about the graph of a line we call that the rise and we divide that by what is the change in X between the points so what is the horizontal distance between the points and we have a word for that it's called run so to get from this point to this point I have to go down two so we would say that the rise is -2 and then I would have to go to the right three units so we say that the run is three something that I should note for you is that when counting the rise if you are going up we consider the rise to be positive and if we're going down we consider the rise to be negative for the run if we go to the right the run is positive but if we go to the left the run is negative now to get this rate of change or slope of -2 over 3 we wouldn't have had to pick these two points linear relationships have a constant rate of change they have a constant slope which means we could have picked any two points and got that same answer let me demonstrate that to you what if I picked the same first point so at -33 but my second point that I picked was way over here at 6 -3 the slope M would be rise over run and the rise I said we would have to go up six units which means the rise is positive six and then we would have to go to the left nine units which means the run is -9 6 over9 could reduce to 2 over3 notice the same value we got when we chose the first two points now that you know how to find the slope of a line from a graph let me erase the second slope calculation that we did and let's focus on calculating slope from a table of values if I were to make a table of values for this linear relationship I would pick some X values and then calculate the Y values and then to calculate the slope I need to figure figure out what is the change in the Y values divided by the change in the X values since we don't have a graph to count the rise and the Run we have to pick two points from our table to algebraically calculate what the rise and run would be I'll pick -33 to be my first point I'll call that X1 y1 and then I'll pick 31 to be my second point I'll call that X2 and Y2 now to find the change in y we could follow the formula Y2 - y1 / X2 - X1 that just finds the difference in y and the difference in X and then divides them if I follow that formula I would have -1 - 3 / 3 - -3 that's -4 over 6 which reduces to that same value we've been getting each time -23 now you know how to find the slope of a line from a graph or from a table of values must know number 3 slope intercept form y = mx + b this is the most common format that you'll see linear relationships represented in in this format m stands for the slope of the line or its rate of change and that's calculated by doing change in y over change in X which we could do in two different ways if we have two points we can find the difference in the two y values and divide it by the difference in the two x values or graphically we can just count the rise and the r and find their quotient the B value in y = mx + b is the Y intercept of the line so let's look at if we have the graph of a line how could we come up with its equation in the form y mx plus b and also if I give you the equation of a line how could we use that equation to sketch the graph of the relationship let's start by looking at the graph of this line if I want to write its equation in the format y = mx + b I'm going to need to know M it's slope and B its Y intercept let's start by finding M the slope of the line which I know I can calculate using the formula rise / run to count the rise and the run I have to pick any two points that I know are on the line and then to get between the two points count the rise which is the vertical distance between the two points so I would have to go up two which means the rise is two and then the horizontal distance between the two points we call that the run and I see you would have to run four units to the right which means the run is four and 2 over 4 of course reduces to a half the B value is just the Y intercept that's just where does it cross the y- AIS and it crosses at two so my B value is two so when I write my linear relationship in the form y = mx + b it would be y = 1 2x + 2 let's now look at what if I gave you the equation of a linear relationship and then I asked you to sketch the graph of this relationship ship we don't have to make a table of values we can just use our understanding of the properties of the yal MX plus b formula the constant that's being added to X is the Y intercept so in this question the Y intercept is four so I can plot a point on the Y AIS at four and the slope is the constant being multiplied by X so our slope which we call M is -2 it's often useful to think of the slope as a fraction so any whole number is over one slope is rise over run so starting at this Y intercept I need to rise -2 which means go down two and then run one which means go right one and then plot a point and then I can just continue that pattern of going down to right one to continue plotting more points to get points on the other side of the Y intercept we can do the opposite this negative sign can go down to the bottom of the fraction and it doesn't change the value of the fraction so that slope of -2 over postive 1 is equivalent to a slope of two over negative 1 both of those have a value of -2 so they're equivalent but the second version would tell me to rise positive2 which means go up to and run Nega 1 which means go left one so if I go up to and left one notice I get another point that is on this linear relationship so there you go now you know how the slope intercept form equation of a linear relationship y = mx plus b is connected to its graph must know number four standard form linear relationships can be represented in another format that looks like this ax + b y + c equal 0 where a B and C are any integers an example of a standard form linear relationship would be the equation 3x + 6 y - 12 = 0 now it should be noted that we could reduce this equation dividing it by 3 to get x + 2 y - 4 = 0 but we'll work with this version the usefulness of writing a linear relationship in this format is that it's easy to calculate both the X and Y intercepts of the linear relationship if I want to know the x intercept graphically that means where does it cross the X AIS well wherever it crosses the X AIS I know its y-coordinate would be zero so to calculate the x intercept we just have to set y equal to Z and solve for x if I do that in the standard form equation that we have the example of I'd have 3x + 6 * 0 - 12 = 0 I would have 3x = 12 divide the 3 I get x = 4 so the x intercept is at the point 4 0 so on my graph I could plot the point 40 if I wanted to calculate the Y intercept graphically that means where does it cross the y- AIS well wherever it crosses the Y AIS at whatever point it does I know its x coordinate is zero so to calculate the y intercept we just set x equal to 0 and then solve for y if I do that I would have 3 * 0 + 6 y - 12 = 0 that leaves me with 6 y = 12 which means Y is equal to 2 so the Y intercept is at 02 now that we have two points that are on this linear relationship we can accurately graph it there's only one line that would cross through these two points and that line would look like this now another important thing for you to know before before I move on from this section would be for you to know how to convert between the slope intercept form of an equation and its standard form equation so what if I give you the equation in the format y = mx + b so for example Y = -23 X + 5 to convert that into standard form we would start by getting rid of any fractions and I could do that by looking at the denominator of the fraction and multiplying the whole equation which means all terms on both sides by that denominator which gives me 3 y = -2x + 15 and now for it to be in proper standard form notice I need to have all the terms on one side of the equation so if I bring everything to the left I would have 2X + 3 y - 15 = 0 what if we start with the standard form equation and we want the slope intercept form of the equation so we might as well work with this standard form equation that we did over there on the left to get that into slope Y intercept form I just have to rearrange it to isolate this y so I'll start by isolating the 6 y by moving the 3x and the -12 to the other side making it -3x + 12 now I'll divide everything by 6 to isolate y which would give me y = -3 6X + 12 6 and that simplifies to Y = 12x + 2 and notice if I look at the graph of that linear relationship this slope intercept form tells me its Y intercept at two and its slope of -1 over2 which means down one right two down one right two there you go you now understand the properties of the standard form equation what it's useful for and how to convert between the formats of a linear relationship must know number five the point slope form of a linear relationship this is the third and final format you might see a linear relationship in which is y - b = m * x - A in this version of the equation m still represents the slope but what's unique about this formula is that a and b the values being subtracted from X and Y represent a point that is on the line let's start by looking at if I give you the graph of a linear relationship how can we generate the equation in point slope form I know I'm going to need to know the slope I'll call it m which I can calculate just by picking any two points on this line and then counting the rise and the Run between those two points to get from my first point to my second point I would have to go down one so my rise is -1 and then I would have to run four so my run is four so in my point slope form equation it's going to look like y - b equals A4 * x - A all I would have to do now is sub in any point that's on my line for a and for B so how about this point right there the point 4 2 that can be my a and my B value so the point slope form equation is y - 2 = a/4 x - 4 that would be the final equation in point slope form let me take one second and actually explain to you why that makes sense we know linear relationships have a constant slope they have a constant rate of change so it wouldn't matter which two points I chose on my line when I did my slope calculation I would always get Nega A4 so let's say I chose some random Point here that has an X and A Y value if I were to find the slope between that XY point and this point right here 42 to calculate the slope I would find the change in the y-coordinates so I would do y - 2 divide that by the change in the x coordinates so I would do x -4 and because all lines have a constant slope I know that that calculation no matter what x y point I plugged into it I would get the answer of A4 so I can set that slope calculation equal to a quarter and now I could just multiply this x - 4 to the other side of the equation and I would have y - 2 = a/4 * x - 4 which is the point slope form of the equation we got over here let's now look at what if I give you the point slope form of the equation and you need to graph the relationship well let me give give you a point slope form equation how about y + 3 = 2 * X - 1 I would start by plotting a point that I know is on the graph of this relationship remember the values that we're subtracting from X and Y so the A and B values are a point that is on the linear relationship so from X I'm subtracting one and from the Y I must be subtracting a -3 to make it look like plus three so I know that a point that is on this line would be the point 1 -3 I'll plot that point and then I need to use my slope to plot more points and what's the slope the slope is the m value which I see two in place of M if I think of two as a fraction that's 2 over one so I have to rise two which means up two and run one which means WR one so from that point I'll go up to right one and plot another point and then continue that pattern to fill the grid if I connect all these points I see my straight line relationship between X and Y there you have it hopefully now you understand the point slope form of the equation as well must know number six find the equation of a line given two points that are on the line Let's do an example let's find the equation of the line that goes through point a which is 2 -3 and point B which is the point -41 let's start by writing the equation in slope intercept form that's yal mx + b form let me start by plotting the two points we have if I want the equation in slope intercept form I need to solve for M and B M represents the slope which when we have two points we can follow the formula it tells us we can do change in y over change in X by finding the difference in the two y-coordinates Y2 - y1 ID by the difference in the two x coordinates if I pick point a to be my first point that's my X1 y1 and if point B is my second Point that's my X2 Y2 if I sub those values into my slope formula I get 1 - -3 / -4 - 2 that's 4 over -6 which would reduce to -2/3 if I want to solve for the B value of this linear relationship well we know M and we know two XY points we just have to pick one of them we could sub those in for MX and Y and then solve for b so I'll write y = mx + b I'll pick point a to be my XY point so the Y value is -3 my M value is -2/3 my x value is 2 and then my only unknown for this relationship is B so I have to solve for b I have -3 = -4/3 plus b to isolate B we'll move the 4/3 to the other side I've got a -3 over 1 + 4/3 = B I'll rewrite that -3 over 1 I want a common denominator so I would multiply top and bottom of it by three making it 9 over3 so I get a B value of -5/3 which means my final equation for this relationship in slope intercept form is y equal -23 x - 5/3 let's connect the points on this graph and see if that matches the equation we found the Y intercept on the graph looks like it's at about 1.67 good and the slope between the points is supposed to be down to right three that's my rise and my run so if I go down to right three yep I get a point down two right three another Point down two right three another Point don't forget we could also write the other two versions of the equation we could write the standard form or the point slope version I'll show you those quickly standard form I would multiply this whole equation by three to get rid of the fractions giving me 3 Y = -2x - 5 I would get everything to one side making it 2x + 3 y + 5 = 0 that would be the standard form equation the point slope equation I would use 2 -3 as my a and my B values which would give me y + 3 equals my slope -23 * x - 2 so when asked to find the equation of a line connecting two points we could have it in any three of these formats all three of those formats are equivalent to each other and all represent that straight line relationship between those two points must know number seven parallel and perpendicular lines parallel lines have the same slope so far to draw two parallel lines they've got the same slopes which mean they would never cross each other let's do an example that involves parallel lines let's find the equation of a line parallel to y = 3x - 5 and goes through the point 71 since I know it's parallel to y = 3x - 5 I know it has the same slope and since y = 3x - 5 is in slope intercept form I know the coefficient of the x 3 is the slope so my parallel slope is three and then I can use the 71 as an XY value that is on my line with that information I could write the equation in the format y = mx + b fairly easily Sub in one for y 7 3 for M 7 for x and then solve for b if I solve this for B I figure out B is -20 which means my equation is y = 3x x - 20 and then I could quickly sketch the graphs of these two lines to show you that they're parallel to each other y = 3x - 5 as a y intercept of5 and a slope of three which means up three right 1 y = 3x - 20 has a y intercept of -20 which I'm not going to be able to fit on the screen here but I do know it goes through the 71 and from there I have a slope of 3 over 1 as well so I can go up 3 right one or I could think of it as -3 over1 which means down three left one notice that those two lines are parallel they would never meet how about perpendicular lines two lines that are perpendicular cross at a 90° angle and a property of their slopes is that they are negative reciprocals of each other and slopes that are negative reciprocals have a product of1 so two perpendicular lines wherever they cross there's going to be a 90° angle I'll explain this word negative reciprocal with an example let's find the equation of a line perpendicular to Y = 1/4 x + 2 and goes through the point -15 now since we did the last equation in slope intercept form let's do this one in point slope form remember point slope form y - b = m * x - A if I want M the slope well I know it's perpendicular to a line that has a slope of 1 4 4 perpendicular slopes are negative reciprocals negative means multiply it by1 so change it sign and reciprocal means do one over the value which means flip the fraction so 1 over 4 I have to flip the fraction to make it 4 over 1 and I have to multiply it by 1 so it becomes -4 over 1 which is just -4 so my perpendicular slope would be -4 the point that the line goes through is 15 so I can make that my a and my B values of my point slope form so the point slope form equation would be y - 5 = -4 x - -1 which I would write as x + 1 and let me graph these two lines so you can see what they look like Y = a/4 X + 2 it has a y intercept of 2 and a slope of up one right four and this equation goes through the point -15 and has a slope of -4 which means down four right one notice those lines are perpendicular to each other they meet at a 90° angle must know number eight horizontal and vertical lines let's start with horizontal lines horizontal lines always are of the format y equals B where B is the Y intercept so for example the line Y = 3 would be the horizontal line through a y intercept of 3 so I plot the Y intercept of 3 and draw a horizontal line through that point I have the line yal 3 why is this defined as yal 3 well if we think about all the points that make up this line what is the y-coordinate of all of those points the y-coordinate of all of those points is three that's why we Define that line is yal 3 how about a vertical line well if for a horizontal line the Y Y coordinates don't change they're constant for a vertical line the x coordinates don't change so they're defined as x equals a where a is the x intercept so an example of this would be x equal -2 if I plot an X intercept of -2 and draw a line that goes through that x intercept that line has the equation x = -2 and why is it xal -2 well if we think about all the points that make up this line the x coordinates of all of those points is -2 the other thing I should show you is what is the slope of horizontal and vertical lines let's start by looking at the horizontal line in fact the slope of all horizontal lines is zero why is it zero well if I were to pick two points on this horizontal line let me label their coordinates remember that when calculating slope we're finding rise over run or change in y over change in X between those two points between those two points the rise is zero right the change in the Y values is 0 - 0 which is 0o and the change in the X values is 5 - 2 so my slope is 0 over 3 and 0 / any number is zero for any horizontal line your rise is going to be zero which is going to make the whole value of the slope be zero what about vertical lines the slope of any vertical line is actually undefined I'll show you that by labeling a couple points on this line and then to find the slope I'll do rise over run or change in y over change in X the change in the Y values is 4 - 2 but the change in the X values there is no change -2 - -2 is zero so my slope is 2 over 0 and anything divided by 0 we say is undefined for any vertical line the change in X or the run is going to be zero so the slope of all vertical lines is undefined must know number nine point of intersection by graphing if I give you the equation of two lines you need to be able to find where the two lines intersect which algebraically means which XY Point satisfies both equations since both of these lines are in slope Y intercept form I know that M represents the slope so the slopes of these lines are a half and four and the B values are the constant terms 2 and5 those represent the Y intercepts I can graph line one by starting plotting the Y intercept at two and then use the slope of a half remember slope is rise over run I can rise one run two and plot a point and then I can follow that pattern to fill the grid and I have the graph of line one now I also have to graph line two and see where they cross and now to graph line two let me do this in a different color I can graph line two by plotting its Y intercept at5 and then using the slope of 4 over one which tells me to rise four so up four and run one which means to the right one so if I go up four and WR one I can plot another point on this line and then continue that pattern and I'll see it actually intersects the other line right there at the point 23 so this right here is my point of intersection algebraically if I were to take this point to three Sub in for X and Y into line one and X and Y into line two it would satisfy both equations and that's the only point that makes left side equal right side for both of those linear relationships must know number 10 finding the point of intersection algebraically let me once again give you two linear equations if you wanted to find the point of intersection of those linear relationships you don't have to to graph it there's actually two methods you can use to find the point of intersection algebraically the methods are called substitution and elimination for the method of substitution I like to start by writing the linear relationships beside each other you then have to isolate one variable in either of the two equations this y would be the easiest to isolate since its coefficient is 1 isolating it would be y = -4x + 7 you then take what you know Y is equal to from the second equation and sub it in for the Y the first equation you're essentially making their y-coordinates be equal that would give me 3x + 5 * Y which I know is equal to -4x + 7 equal to 1 so I've made their y-coordinates be equal and now I'm going to solve for what value of x allows that to be true I can distribute the five to both terms 3x - 20x + 35 = 1 I have -7x = -34 divide -7 I figure out x = 2 I can now take that answer I got for x sub it into this equation for x and then solve for y if I do that I'd have y = -4 * 2 + 78 + 7 is1 so the point 21 is the point of intersection of those two linear relationships or algebraically speaking these are the X and Y values that s satisfy both of those equations the other method that you could use for finding that same point of intersection is the method of elimination to start elimination you start by writing the equations on top of each other and now you want to make the coefficients of either the X's or the Y's have the same absolute value I could make the coefficients of the Y's be the same if I multiplied this bottom equation by five the top equation I can leave the same and now what I do with these two equations is I need to subtract their like ter terms to generate a new equation that will allow me to solve for x with the Y variable being eliminated If I subtract the like terms 3x - 20x is -7x 5 y - 5 Y is 0 y the Y's are eliminated and 1 - 35 is -34 if I solve this equation for x I get X is 2 then we can take this answer for x and sub it back into either original equation for x and solve for y if I sub it back into the first equation I'd have 3 * 2 + 5 y = 1 5 y = -5 which means Y is -1 so my point of intersection again I would get as 21 hopefully this video helped you understand linear relationships better let me know what you want a top 10 of next