Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Horizontal lines are lines of the form Y equals some number. Now what this really means is that all of our X's don't matter when we graph this line, that the Y is always some number. So if we look at this graph, let's put in a line right here. If we thought randomly about what two points on this line are 123456, so we could say the .16 is on this line. We could say the point -3 six is on this line. So every point on this line has AY value of six. It doesn't matter what the X value is. Now, to find a slope, we're going to take the difference of the YS over the difference of the XS. And in this case we get 0 / 4 and 0 divided by any numbers, just zero. So horizontal lines are of the form Y equal a with the slope of 0, a vertical line. Vertical lines are going to go up and down. So let's put one in, let's say right here just randomly. So in this example, all of our X's have the same value. We went over three, so here's a .32 or up here is a .3 12345. We're down here three -1. So it doesn't matter what our Y value is this time. All of our X's are three. So the vertical line is going to be the form X equals some number, let's call it C Doesn't matter, it's just a variable or constant. So if X = C. Now to find the slope, remember it's the change of YS, so 5 -, 2 over the change of X's. This case is 3 / 0. Well, we can never divide by zero, so the slope is undefined. Now be careful, undefined is not the same thing as no slope. No slope would actually mean the slope doesn't exist. But the slope does exist, it's just an undefined number. Intercepts to do an intercept. AY intercept says X is going to be 0 when we have our Y value. So for AY intercept we set X equal to 0 and solve for Y. So the Y intercept is a point and it's a point somewhere along the Y axis. If we think about it being a point, this point didn't go right or left any, but we went up or down some. The X intercept, same concept, except now we're going to be on the X axis, so it's some point on the X axis. So in this case, we went right or left sum, but we didn't go up or down any. So in this case, we'd set Y equal to 0 and solve for X. Now if we know our X intercept and our Y intercept, we could actually connect those two and we could graph are aligned using intercepts. Standard form of a line is the form ax plus B y = C where AB and C are integers IE not fractions or decimals. A&B are on the same side and A is greater than or equal to 0. So we have slope intercept form and the slope intercept form was Y equal MX plus B where M was our slope and our .0 B was our Y intercept. And now we have another form AX plus B y = C where AB and C are integers. A&B are on the same side a greater than or equal to 0. Sometimes this is thought of as the pretty form of a line. So oh, we needed to talk about parallel and perpendicular. So parallel is when the two slopes are the same. So parallel lines have the same slopes. Perpendicular lines have slopes that are negative reciprocals. An easy way to do this one is if the two slopes multiply to give -1. So if M1 times M 2 = -1. Now, the exception to this is that a vertical line and a horizontal line are also perpendicular. So perpendicular slopes are negative slopes that are negative reciprocals. The two multiply to give us -1 or it's a vertical line and a horizontal line. The word perpendicular means that the two lines are going to form a right angle. So if you look here, if we just look at the two axis, we have a right angle. That notation is frequently a part of a square to show that right angle. So our first examples 5 -, 3 Y equals seven. We want to graph this and if we solved for Y to put it in Y equal form, we'd have -3 Y equaling 7 - 5 is 2. So this is a line that says Y equal -2 thirds. Well, AY equal line we know is really just a horizontal line. So the slope here is 0 and our line is Y equal negative 2/3. This next one, if we solve for X we get 2X subtract 3 so -11 X equal -11 halves. So if we draw our graph -11 halves, remember is really two goes into 11/5 full times with one leftover. So we're going to be out here and this is going to be an X equals. So it's a vertical line. So the slope is undefined. Now our next one is, are these two parallel, perpendicular or neither? So we're going to put them in Y equal form to find it. So the first one, we're going to subtract the 2X to the other side, so Y equal negative two X + 4. That tells me my slope is -2. It didn't ask for it, but it doesn't hurt to practice. 04 would be our Y intercept. So for this next equation, let's solve for Y. Again, we're going to add the 6X over. To make it a positive. We're going to divide by -3, so we get -2 X -6. So our slope here is -2 our Y intercept zero -6. So the fact that the two slopes of the two equations are the same means that these two lines are parallel. We know that they're parallel and not the same line because the Y intercepts were different. So if we were going to graph these 04, we'd go up four, and our slope being -2 -2 we could think of as down to into the right one, down to into the right one. So that would be that top equation. The second equation would be -6 and then we would go down to into the right one. I could also think of it as up two into the left one. If I drew that in, and if I drew a little more accurately, it'd be easier to see. But those would indeed be parallel. Same exact directions. We want to know parallel, perpendicular, neither. So we're going to take this first one. We're going to solve for Y, subtract the 4X, divide by 3. So on this first one we get a slope of -4 thirds and we get an intercept of 07 thirds. On the second one solve so we get -8 Y negative six X + 2 divide by -8 Y equal. I'm going to go ahead and simplify as I go 3 fourths X -, 1/4. So a slope here of 3/4 and AY intercept of 0 negative 1/4. We can see now that 3/4 and -4 thirds, they're what's called negative reciprocals of each other. Or if I multiply them together, they do indeed give me one -1. So these are perpendicular, they're negative reciprocals. So this last example for this type, once again we're going to solve for Y SO5Y negative X + 9 Y equal -1 fifth X + 9 fifths. So a slope of -1 fifth and 09 fifths for Y intercept. This next one -4 Y equal negative two X + 5 Y equal 1 half X - 5 fourths slope is a half zero -5 fourths. These slopes are not the same, hence not parallel. They're not negative reciprocals, hence not perpendicular. So these are neither parallel or perpendicular. This next one, we want to solve this by graphing. To solve this by graphing, what we're going to do is we're going to actually let the left side equal Y and graph this line, and then we're going to let the right side equal Y again and graph that line. So if I graph Y equal to X + 3, we would think about going up three because that's our Y intercept and our slope is 2. Remember, 2 could be thought of as 2 / 1, and our Y intercept was 03. So from 03 we're going to go up to and to the right one, up to and to the right one, or I could go down to and to the left one if I wanted. Now this Y equal 5 over here, Y equal 5 is really, really just a horizontal line 12345. So we're going to put in our horizontal line here, and the way to solve it is to look for where the two intersect. What X value is where these two intersect. They have two intersect when X = 1. So if X equal 1, that's our solution. That's where these two lines intersect. We could check it does 2 * 1 + 3 really equal 5? And the answer is yes. So that's how to solve an equation graphically instead of algebraically. Take the left side, set it equal to Y. Take the right side, set it equal to Y. Graph the two lines and look for the X value where the two intersect. This next example we want to solve, we want to graph using the X intercepts and Y intercepts. So X intercept, Let's do the X intercept. Set Y equal to 0, so four X + 3 * 0 equal 24, four X equal 24 X equals 6. This is a point. It's the point when X IS6Y was 0. So then find the Y intercept. One method, and there are many, is to set X equal to 0. So 4 * 0 plus 3Y equal 24. Three Y is 24, Y equal 8. This once again is a point. It's the point when X was 0, Y was eight. Now a different way to find the Y intercept would be to solve for Y. So we could have taken 3 Y equaling negative four X + 24 and then divide by that three. And we could have seen that this number here without the X is our Y intercept. So to graph this 60, 8:08 connect the two points, there's our line. If we had put it in Y equal MX plus B form 08 was our intercept. Our slope was -4 / 3. So starting at 08, if I went down four and to the right three, down four to the right three, I can see that I got the same line graphing it in two different methods. So rewrite in slope intercept form. We want to solve for Y. SO7Y would equal negative two, X + 8 Y would equal -2 sevenths, X + 8 sevenths. So our slope would be -2 sevenths. Our Y intercept would be 08 sevenths. Now we're going to let you try 1. This one we're going to rewrite in standard form. So if we're going to rewrite it in standard form, we're going to multiply everything through by the fraction first. So 6Y equal five X + 24. I need the X's and Y's on the same side, and I need the X to be positive. So we're going to subtract the six Y so that the X's and Y's are on the same side and the X is positive. And then I need the constant or the number without an X or Y to be on the other side. So we're going to subtract 24 now. And typically we write it so that the variables come first. So our final is going to be 5 X -6 Y equal -24. Thank you and have a wonderful day.