in this video we're going to do a review of linear equations it's especially for those of you who have a test that you're studying for so let's begin by writing down some notes there are three forms in which you can write a linear equation the first one is known as the slope intercept form in the slope intercept form the linear equation is written this way y is equal to mx plus b m represents the slope which we'll talk about later and b that represents the y-intercept we'll also talk more about that later as well but for now you want to write this equation so this is the slope-intercept form of a linear equation now the next form that you want to be familiar with is the standard form so to write a linear equation in standard form this is how it's going to look like it's a x plus b y is equal to c a b and c are simply coefficients x and y are the variables but when written in that form it's called the standard form the next form that you need to be familiar with is this one the point-slope form of a linear equation it's y minus y one is equal to m times x minus x one as the name implies this equation can tell you the slope and the point the slope is the value of m so whatever number you see here and that's the slope the point is x1 comma y1 now let's talk about the slope the slope is equal to the rise divided by the run so let's say if you have a linear equation that is rising the slope is going to be positive and let's say you have two points on this line now to go from the first point to the second point let's say it takes you have to travel up by four units this is not drawn to scale by the way this is just an illustration so let's say you travel up four units and then you travel three units to the right so in this case your rise is four your run is three so rise over run the slope would be four over three and because it's going up the line is going up the slope is going to be positive now here's another example let's say to go from this point to that point we need to go down three units so the rise is negative because we're going down so let's say negative three units let's say we have a run of positive five it's positive because we're moving to the right the run should always be positive now for this one the slope is going to be rise over run the rise is negative 3 the run is 5 so it's going to be negative 3 over 5. so because the line is going down the slope is negative so that's a quick and simple way to calculate the slope using the rise over run method now whenever a line it goes up at a 45 degree angle the slope of that line is going to be one if it goes up like this it's about one half and if it goes up even steeper let's say like this this would be a slope of two so this line is very steep compared to the other ones now if the line is horizontal the slope is going to be zero if it goes down at a 45 degree angle like this the slope is negative one here it's about negative a half and here negative two so any time the slope increases i mean anytime the line increases the slope is going to be positive if the graph is going down the slope is negative and for any horizontal line the slope is zero so if we have a line that's going to the right or to the left the slope is zero for a vertical line the slope is undefined as the line becomes more vertical the slope increases eventually it can go up to infinity and at some point it will be undefined so just know that so if you have a vertical line the slope is undefined for a horizontal line the slope is zero now you can calculate the slope of a line if you know the two points so let's say if the first point is x one comma y one and the second point is x2 y2 the slope of the line is going to be y2 minus y1 divided by x2 minus x1 so here's an example let's say the first point is two comma five and the second point is let's say five fourteen go ahead and calculate the slope so y2 is going to be 14 and let's replace y1 with five x2 is five x one is two fourteen minus five is nine five minus two is three nine divided by three is three so the slope of the line that connects these two points is equal to 3. now let's talk about x and y-intercepts what is an x-intercept and what is a y-intercept what do you think the answer to that question is an x-intercept is a point but it's a specific point the x-intercept is the point where y is equal to zero so let's say you have the point three comma zero at this point x is three y is zero this particular point is an x intercept because the y value is zero so the x-intercept is any value of x when y is zero another example of an x-intercept is the point negative five zero so the x-intercept in this case will be negative five if we have the point two comma zero the x intercept is two so any point where the y value is zero the x value is the x intercept for the y-intercept the situation is reverse the y-intercept is the y-coordinate of a point when x is 0. so let's say if we have the point 0 4 the y-intercept is 4. so remember when dealing with linear equations the y-intercept is also equal to b so we would say that b is 4. here's another example of a y-intercept let's say the point zero negative three so x is zero y is negative three the y intercept is negative three so we can say that b is equal to negative three now let's summarize what we've just learned the x intercept is the x coordinate of a point that contain a y value of zero so as we see here this point has a y value of zero the x intercept is the x-coordinate of that point so x is negative five the y-intercept is the y-coordinate of a point that has an x-value of zero so for this point the x value is 0 but the y intercept is the y coordinate of that point so it's y equals 4 or b equals 4. so that's the basics of the x and y intercepts so here's an example problem for you consider these four points the point two comma five negative three comma zero one comma two and zero comma six given these four points identify the x and y-intercepts so this point here has a y value of zero therefore that point represents the x-intercept the x-intercept is specifically the x-coordinate of that point so the x-intercept we could say it's x equals negative three but it you can also say that the x-intercept is the point negative three comma zero you can describe it both ways the y-intercept is the point where x is zero so this would be the y intercept now you can say the y intercept is y equals six or you could say it's a b equals six now the next thing that we need to review are parallel lines and perpendicular lines so what's the difference between parallel lines and perpendicular lines what would you say and how do their slopes relate to each other parallel lines they travel in the same direction let's call this line one and line two let's say the slope of line one has a value of two if the slope of line one has a value of two and if line two is parallel to line one the slope of line two will be the same it will also be two what you need to know is that parallel lines they have the same slope so m1 is going to be equal to m2 you can also describe the relationship between two lines that are parallel using the symbol if you see this double vertical line it means that the two lines are parallel so this is saying l1 is parallel to l2 but for linear equations if you have a test we need to know is that parallel lines they have the same slope now let's talk about perpendicular lines let's say we have this line which we'll call l1 and then this line l2 perpendicular lines they intersect at right angles that is at 90 degrees the slope of the perpendicular line is the negative reciprocal of the original line so let's say the slope of line one let's say it's positive three over four the slope of line two is going to be the negative reciprocal so you've got to change the sign from positive to negative and you've got to flip the fraction so it's going to be negative 4 over 3. so m1 is going to be negative 1 over m2 so that's the relationship between the slopes of two perpendicular lines so we could say l1 is perpendicular this is the symbol for perpendicular l1 is perpendicular to l2 you can see these two lines meet at right angles so that's how you can describe two perpendicular lines so remember the slopes are negative reciprocals of each other now let's work on some example problems let's say that line one is parallel to line two and let's say that you're given the slope of line one let's say that the slope of line one is negative three what is the slope of line two if they're parallel the slope of line one is equal to the slope of line two therefore the slope of line two will be negative three now let's say that line one is perpendicular to line two and let's say you're given the slope of line one let's say it's negative four over seven what is the slope of line two the slope of line two is going to be the negative reciprocal of the slope of line one so the first thing you need to do is change the sign from negative to positive and then you need to flip the fraction from four over seven to seven over four so that's going to be the slope of the line that's perpendicular to the first line now let's talk about how we can graph equations in slope intercept form so let's say we have the equation y is equal to 2x minus 4. how can we graph this equation so first let's put in some marks on a graph feel free to try this example if you want to so the first thing we need to identify is the slope and the y intercept so this is in y equals mx plus b form it's in slope intercept form so we can see that the slope is equal to two and we can see that the y-intercept is negative four with this information we have everything that we need in order to graph this function so here's negative four let's go ahead and plot the y-intercept and then from the y-intercept we can get the second point by using the slope so the slope is 2 which means that it's 2 over 1. so the rise is two the run is one so to get the next point we're going to go up two units and then travel one unit to the right so that will give us the point one negative two so we have an x value of one and a y value of negative two now let's go up two and over one again so we get the next point which is two comma zero so that's an x-intercept the x-intercept is the point of the graph that touches the x-axis because on the x-axis y is zero the y-intercept touches the y-axis so this point is zero negative four it's the y-intercept because x is zero and this is the x-intercept because y is zero now all you need is two points in order to graph a line so we can add more points but we can just connect these points with a straight line so that's how we can graph this equation in slope intercept form now let's say we have this one y is equal to negative three over four x plus five how can we graph this equation feel free to pause the video if you want to try it so first let's identify the slope and the y-intercept so we can see that the slope is negative 3 over 4 is the number in front of x the y-intercept is 5. so first we're gonna plot the y-intercept the y-intercept has the point zero negative five x is zero y is negative five so it's on the y axis this is the x axis this is the y axis so now that we have the first point the y intercept let's use the slope to get the next point so starting from the y-intercept we have a rise of negative three and a run of four so this is the rise this is the run so as we travel down three and over four it's going to take us to this point so that is four on the x-axis two on the y-axis now all we need is two points to graph a line so now we can just draw a line that connects those two points and so that's how we can graph that particular linear function now let's move on to the next example so this time we're going to graph a linear function in standard form so we have 3x minus 2y is equal to 6. so it's an ax plus by equals c format how can we graph a linear equation in standard form what do you think we need to do one of the most simplest techniques that you can use is to find the x and the y intercepts to find the x intercept replace y with zero negative two times zero is zero so we get just three x is equal to six solving for x we can divide both sides by three and so we get x is equal to six divided by three which is two so the x intercept is two comma zero x is two y is zero since we replaced y with zero now let's find the y intercept to find the y intercept replace x with zero three times zero is zero so we're just going to get negative two y is equal to six dividing both sides by negative two we get that y is six divided by negative two which is negative three so the y intercept is going to be 0 negative 3. so what we're going to do is we're going to plot the x intercept which is here it's 2 comma zero and then let's plot the widest up the y intercept is zero negative three so now let's connect these two points with a straight line and that's all you need to do in order to graph a linear equation in standard form let's try another example so let's say we have 4x plus 3y is equal to 12. go ahead and graph that linear equation so let's find the x-intercept let's replace y with zero so we're going to get 4x is equal to 12 and then dividing both sides by four we get x is 12 over four which is three so the x intercept is three comma zero now let's get the whiteness up so this time let's replace x with zero four times zero is zero so we just get three y is equal to twelve divide both sides by three so y is twelve divided by three which is four so we get the point zero comma four so the x-intercept is at three the y-intercept is at four and then we just need to connect those two points with a straight line so that's how we can graph the linear equation in standard form 4x plus 3y is equal to 12. now what about an equation that is in point slope form let's say we have y minus 3 is equal to 2 times x minus 2. how can we graph an equation in that form feel free to try that problem so this is in y minus y1 is equal to m times x minus x1 form that's the point slope form in that form we could find the point and the slope so here's the slope the slope is 2. now we can also find a point through which the line passes through and that point is x1 y1 so what's x1 and what's y1 notice that these two negative signs are the same therefore 1 has to be positive 2 because those negative signs already there so x 1 is positive 2 y 1 is 3 without the negative sign so when you see x minus 2 the point is going to be 2. change the negative sign into a positive sign if you see y minus 3 the y coordinate is positive 3. so with this information we can graph it we have a point and a slope so let's plot the point two three so here is two three the x value is two the y value is three and then we could use the slope to get the next point the slope is two so we have a rise of two and a run of one so we can go up two and over one to get the next point so that's going to be three comma five and we can go backwards let's say if we go one to the left we need to go down to because there's not much space in the right side of this graph so that's how we can graph a linear equation in point slope form for the sake of practice let's do one more example so let's say we have the linear equation y plus four is equal to negative three over two times x plus one so go ahead and graph that linear equation so let's begin by identifying the slope the slope is negative three over two now what's the point here we have x plus one the x coordinate is going to be negative one simply reverse positive one to negative one here we have y plus four the y coordinate will be negative four so now we have a point and a slope that's all we need in order to graph this function so the first point is that negative one negative four which is here the x value is negative one the y value is negative four and then to get the next point the slope is negative three over two so we need to go down three and over two but it looks like we're out of space so we're going to go backwards that is we're going to go up three and then two to the left so up three two to the left that still gives us the same slope that's a rise of three a run of negative two which is still negative three over two so sometimes you may need to go backwards like in this problem so if we go up three and over two we should be at this point and this point is at negative three comma negative one now at this point we can go ahead and draw a line between these two points so that's a rough sketch of the graph that corresponds to this linear equation now what would you do to graph this equation let's say y is equal to 3 how can you graph that whenever y is equal to a constant number what you're going to get is a horizontal line in this case a horizontal line at three so if we wanted to graph y is equal to negative two we would simply draw a horizontal line at negative two along the y axis so whenever y is equal to a constant you're going to get a horizontal line and the slope of that line is going to be 0. now what if we wanted to graph x is equal to four in this case we're going to have a vertical line at x equal four so this line will contain all points with the x coordinate x equals four the slope of that line is undefined if we want to graph x is equal to negative 3 it's simply going to be a vertical line touching all points with the x coordinate negative 3. so that's how we can graph that now let's work on some multiple choice and free response practice problems that's going to help you to review for the tests if you're studying for one number one which of the following graphs correspond to the equation y is equal to two x minus three so this equation is in slope intercept form now there's two things we need to focus on we need to identify the slope and the y-intercept the slope is the number in front of x so therefore the slope is equal to 2. the y-intercept is the constant that you see next to the 2x so the y-intercept which is b is negative 3. so let's identify the graph with the correct y-intercept if we look at answer choice a the graph touches the y-axis at positive three therefore answer choice a is not correct looking at b c and d the graph touches it at negative three so far c b and d are okay now let's look at the slope the first thing we want to notice is that the slope is positive a positive slope means that the function is increasing a negative slope means that it's decreasing and for a horizontal line the slope is zero so because the slope is positive the graph should be going up therefore we can delete d because it's going down graph d has a negative slope now between b and c what's the difference well let's look at c as we travel one unit to the right notice that the graph goes up by three so the slope is three now let's look at b as we travel one unit to the right notice that the graph goes up by two which gives us a slope of two so b is the right answer it has a y-intercept of negative three and a slope of two you