in this video we're going to focus on writing equations of lines that are parallel and perpendicular to a given line but pass them to a point so let's say if we have the equation Y is equal to 2x + 1 and we want to write the equation of a line that's parallel to the line y = 2x + 1 and that passes through the point 1A 5 how can we do it well for this particular example let's use the slope intercept form which is y = mx + b m represents the slope which is rise over run and B is the Y intercept so for this particular example you need to realize that the slope of the other line is two and parallel lines have the same slope so the slope of the line that we're trying to find also has a value of two so let's replace M with two now what we need to solve for is B the Y intercept so let's plug in X and Y so let's replace y with five and x with one so we have 5 is equal to 2 + B so to solve for b we need to subtract two from both sides so 5 - 2 is 3 so B is equal to 3 now once you have the value of B go back to the original equation in slope intercept form and replace B and M but leave X and Y alone so then the equation that we're looking for is y is equal to 2x + 3 and that's the answer go ahead and try this example let's say that Y is equal to 3x - 4 and you want to write the equation of the line that's parallel to this line and that passes through the point 6A 4 so feel free to pause the video work through this example and then unpause it to see if you have the right answer so let's begin so starting with the equation Y is equal to MX plus b let's replace M with the slope and let's solve for b and let's plug in uh X which is six and four for y so 4 is = to 3x + Well 4 is equal to 3 * 6 plus b so 3 * 6 is 18 and if we subtract 18 from both sides we could see that the value of B is going to be 4 - 18 which is -14 so now we can write the equation of the line so it's going to be Y is equal to let's plug in m and b so m is 3 so Y is = 3x - 14 and so that's how you can write the equation in slope intercept form try this example Y is = 3 over 2 x + 3 and this time let's say the point is 43 so go ahead and write the equation of the line that is parallel to this line and passes through the point 4 comma -3 and feel free to write the equation in point slope form so what I'm going to do is use the point slope equation which is y- y1 is equal to m x - X1 so I'm going to write the equation in point slope form first and then I'm going to convert it to slope intercept form so this is X1 and this is going to be y y 1 and the slope is still 3 over2 it's the number in front of X so let's plug in y1 let's replace it with -3 and let's substitute M for 3 over2 and X1 is 4 now whenever you have two negative signs next to each other it's going to change into a positive sign so what we now have is y + 3 and let's distribute the three halves to x - 4 3 over2 * X that's going to be 3 over 2X and what's 3 over2 * 4 so 3es * 4 1 3 * 4 is 12 12 divid 2 is 6 so this is going to be -6 so the next thing we need to do is well this is the answer in point slow form by the way here it is I should have mentioned that it's y + 3 = 32x - 4 that's the equation in point slope form now once you distribute the 3 over2 you're going to convert it into slope incept form so to finish it all I need to do is subtract both sides by three so the answer in slope intercept form is y = to 32x - 9 if you use the method that we used in the first two examples this is the answer you should get but in point slope form it should be like this but it's just going to be y + 3 equals to what's on the right side now try this one let's say if Y is equal to 5x - 3 and we have the point 21 so here are the instructions let's you want to write the equation of the line that is perpendicular to Y = 5x - 3 but passes through the point 2 comma 1 what do you need to do the first thing you need to do is you need to change the slope for a parallel line the slope so the same but if it's perpendicular the slope is going to change from 5 over 1 to 1 over 5 with a negative sign so you need to flip the fraction and you need to change the positive sign to a negative sign now all you need to write the equation of the line is the slope and a point that passes through the line once you have those two things you can write the equation of the line so let's do it but let's do it in point slope form and then we'll convert it to slope intercept form so using equation y - y1 is = to M * x - X1 so X1 is going to be 2 y1 is1 so y - -1 is equal to -1 over 5 * x - 2 so on the left we have y + 1 and on the right we have -1 over 5 xus 2 so this is the answer in point slope form but now let's go ahead and get the answer in slope intercept form so let's distribute the -1 over 5 so it's going to be y + 1 is equal to -1 over 5x and then -5 * -2 that's going to be positive 2 over 5 and now let's subtract both sides by one so on the left side all we have is y and on the right side we have -5x + 2 over 5 now -1 is the same as-5 over 5 if we uh convert it so we can get common denominators 5 over 5 when you divide them you get negative 1 and so since the denominator is the same we can now combine the numerators of the fractions so 2 over 5 - 5 over 5 well 2 - 5 is-3 so this is going to be-3 5 so this is the equation in slope intercept form so what about this one y is equal to 3 over 4x - 1 and let's say it passes through the point 8 -3 so and let's say it's perpendicular to the line 3 over4 X -1 how can you write the equation of the new line so this let's use the slope intercept equation which is yal mx + b so we got to find a slope it's going to change from 3 over 4 to we got to flip the fraction so it's going to be 4/3 and change a sign from positive to negative so this is the slope of the perpendicular line which we're trying to find so and let's plug in X for 8 y for -3 and let's isolate or let's solve for b so -3 is = to -43 * 8 plus the Y intercept which is B so to get rid of the fraction let's multiply everything by three whatever this denominator is so 3 * -3 that's going to be -9 and 3 * 4/3 * 8 the threes are going to cancel so you're just going to get -4 * 8 and then 3 * B that's going to be 3B so -4 * 8 that's -32 and let's make some space now let's add 32 to both sides so on the right side we just have 3B and on the left side 32 - 99 is 23 so if we divide both sides by three we can get the value of B so B is going to be equal to 23 over 3 and so now we have the equation it's going to be Y which is equal to the slope -4/3 X plus the Y intercept 23 over 3 so this is the answer so here's the final example for today Y is = -23 x + 4 with the point 6 -2 so go ahead and find the equation or write the equation of the line that's perpendicular to the line -23 x + 4 and passes through the point 6 -2 so I'm going to use the slope intercept form again so Y is equal to mx + b so the slope is going to change from 2/3 it's a positive 3 over2 if we're looking for the perpendicular line and so let's plug in 6 for x and -2 for y so -2 is equal to 3 over2 that's the new slope times the value of x which is 6 plus b so what's three halves time 6 6 ID two that's three 3 * 3 is 9 so 3 * 6 is 9 and to isolate B we need to subtract both sides by 9 so -2 - 9 is1 so B is equal to1 so now all we got to do to write the answer is plug in m and b so m is 3 over2 and b is1 so this is the equation of the line in slope intercept form