Video Transcript
In this video, we’re going to look at how to find the equation of a straight line in various forms when we are given two specific pieces of information, the slope of the line and the point that lies on the line. First, to remind of the different formats that we can use when expressing the equation of a line. The first format is referred to as a slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. The letters 𝑚 and 𝑐 have specific meanings here. 𝑚 represents the slope of the line. So for every one unit that you move across, the line would move 𝑚 units up. If 𝑚 is negative, then for every one unit you move across, the line would move 𝑚 units down. 𝑐 represents the 𝑦-intercept, which is the value at which the line intercepts the 𝑦-axis.
The second format that is often requested is the point-slope form of a line, 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one. Here, 𝑚 refers to the slope of the line in the way we just described. 𝑥 one, 𝑦 one refer to the coordinates of a point on the line. This is a general point anywhere along the length of the line, not specifically the 𝑦-intercept. In this video, we will find the equations of lines in both of these two forms.
A line 𝐿 has a slope of three and passes through the point three, four. Work out the equation of the line, giving your answer in the form 𝑦 equals 𝑚𝑥 plus 𝑐.
So we’re given two pieces of information in the question here. We’re told first of all that the line has a slope of three. And we’re also told it passes through the point three, four. We’re asked to give our answer in the form 𝑦 equals 𝑚𝑥 plus 𝑐, which is slope-intercept form. So let’s begin. We need to work out the values of 𝑚 and 𝑐, the slope and the 𝑦-intercept. We’re actually given the slope within the question. We’re told that it’s three. So straightaway, we can write down part of the equation of the line. It’s 𝑦 equals three 𝑥 plus 𝑐.
We now need to work out the value of the 𝑦-intercept 𝑐. So we’re told that this line passes through the point three, four. This means that when 𝑥 is equal to three, 𝑦 is equal to four. And therefore, we have a pair of values for 𝑥 and 𝑦 that we can substitute into this equation in order to find the value of 𝑐. So I substitute the value four for 𝑦 and the value three for 𝑥. This gives me an equation that I can solve. I now have that four is equal to nine plus 𝑐. And if I subtract nine from both sides of this equation, I now have that 𝑐 is equal to negative five. Finally, I just need to substitute the value of 𝑐 into my equation for the line. And we have our answer to the problem. The equation of the line is 𝑦 equals three 𝑥 minus five.
Find, in point-slope form, the equation of the line with slope two-sevenths that passes through the point A, one, negative 10.
So we’re given two pieces of information about this line. We’re told that it has a slope of two-sevenths. And we’re told that the point with coordinates one, negative 10 lies on this line. We’re also asked to give our answer in point-slope form. That is the form 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one. So to do this, we just need to recall what 𝑚, 𝑥 one, and 𝑦 one represent in this form of the equation of the line. 𝑚 represents the slope of the line. And we’re told that this is equal to two-sevenths. Therefore, we can substitute the value of two-sevenths for 𝑚. 𝑥 one and 𝑦 one represent the coordinates of a point on the line. And therefore, we can substitute the value one for 𝑥 one and the value negative 10 for 𝑦 one. So we have 𝑦 minus negative 10 is equal to two-sevenths 𝑥 minus one. We don’t need to do much more work with this, as this is the format that we’re requested to give our answer in. But we can just tidy up the left-hand side a little. 𝑦 minus negative 10 just simplifies to 𝑦 plus 10.
So our answer to the problem is that, in point-slope form, the equation of this line is 𝑦 plus 10 is equal to two-sevenths of 𝑥 minus one.
What is the equation of the line with 𝑥-intercept negative three and 𝑦-intercept four?
So we have two pieces of information about this line. We know both its 𝑥-intercept and its 𝑦-intercept. Therefore, the line looks something like this. We haven’t been asked to give our answer in a particular format. So I’m going to choose to use slope-intercept form. 𝑦 equals 𝑚𝑥 plus 𝑐. Now straightaway, I can fill in some of the information here. Remember, 𝑐 represents the 𝑦-intercept of the line. And I’m told that this is equal to four. Therefore, the equation of this line is of the form 𝑦 equals 𝑚𝑥 plus four.
Next, I need to calculate the value of 𝑚, the slope of this line. The slope of line can be calculated using change in 𝑦 over change in 𝑥. Or 𝑦 one minus 𝑦 two over 𝑥 one minus 𝑥 two, where 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two are the coordinates of two points on the line. Now I do have the coordinates of two points on the line. If the 𝑥-intercept of the line is negative three, then this means that the point with coordinates negative three, zero is on the line. If the 𝑦-intercept is four, then the point with coordinates zero, four is also on the line. So I have the two points that I need in order to calculate this slope. It doesn’t matter which of these two points I choose to be 𝑥 one, 𝑦 one and which I choose to be 𝑥 two, 𝑦 two. As long as I’m consistent about the order in which I subtract the 𝑦-values and the 𝑥-values. I’ve chosen to allocate the points in this order. And therefore, my slope is four minus zero over zero minus negative three. This gives me a slope of four-thirds.
So now I can substitute the value of this slope into the equation of my line. I have then that 𝑦 is equal to four-thirds 𝑥 plus four. Now sometimes this might be an acceptable form in which to leave my answer. But given that that slope is fractional, I would prefer this answer not to have any fractions within it. So in order to deal with this, I’m gonna choose to multiply through the equation by three. So multiplying every term in this equation by three, I have then that three 𝑦 is equal to four 𝑥 plus 12. Don’t forget to multiply that 𝑦-intercept by three as well. That’s a common error. Finally, I’m going to choose to group the 𝑦 and 𝑥 terms together on the same side of the equation. So I’m gonna subtract four 𝑥 from both sides. This gives me my final answer to the problem in the format I’m gonna choose to leave it in. Three 𝑦 minus four 𝑥 is equal to 12.
Write the equation represented by the graph shown. Give your answer in the form 𝑎𝑥 plus 𝑏𝑦 is equal to 𝑐.
So I’ve been given a diagram of the line that I’m looking to find the equation of. I haven’t been asked to give the equation in a particular form. So I’m going to choose to work using point-slope form. In order to find the equation of the line then, I need to know two things. I need to know the slope of the line, 𝑚. And I need to know the coordinates of one point on the line, 𝑥 one, 𝑦 one. So let’s look at this graph more closely. I can see straightaway that the 𝑦-intercept of the line is negative nine. Which means that the line passes through the point with coordinates zero, negative nine. If I look along the length of the line, I can also see that it passes through this point here. This point has the coordinates four, negative six. So I can use these two points together in order to work out the slope of the line.
So the slope of the line is the change in 𝑦 divided by the change in 𝑥. Or you may refer to this perhaps as rise over run. Using the two points that we can see here on this line, this gives me a calculation for the slope of negative six minus negative nine over four minus zero. This calculation simplifies to three over four. So I have the slope of the line. And I can substitute it into my equation. I now have 𝑦 minus 𝑦 one is equal to three-quarters 𝑥 minus 𝑥 one. Now I can choose to use either of these two points that I know on the line as 𝑥 one, 𝑦 one. For simplicity, I’m going to use the point zero, negative nine as one of the values within that coordinate is zero. So substituting the values for 𝑥 one and 𝑦 one, I now have 𝑦 minus negative nine is equal to three-quarters of 𝑥 minus zero. This simplifies to 𝑦 plus nine is equal to three-quarters of 𝑥.
Now if I look back at the question, I can see that I’m requested to give my answer in the form 𝑎𝑥 plus 𝑏𝑦 is equal to 𝑐. And although the question doesn’t state this, it’s usual that 𝑎, 𝑏, and 𝑐 would represent integers. So in order to get my answer into this form, the first thing I need to do is to multiply everything in the equation by four. As that will remove the four from the denominator of the fraction on the right-hand side. In doing this, I now have the equation of the line as four 𝑦 plus 36 is equal to three 𝑥. However, the requested form has the 𝑥 and 𝑦 terms both on the same side of the equation. So the final step is I’m going to subtract four 𝑦 from both sides. This gives me then my answer to the problem in the requested form. The equation of the line is three 𝑥 minus four 𝑦 is equal to 36.
Write the equation represented by the graph shown. Give your answer in the form 𝑦 equals 𝑚𝑥 plus 𝑏.
So this is represented slightly differently from how we usually see the slope-intercept form of a line. We usually see it as 𝑦 equals 𝑚𝑥 plus 𝑐. But the letter makes no difference. Whether it’s 𝑏 or 𝑐, it just refers to the 𝑦-intercept of the straight line. From looking at the diagram, we can see that the 𝑦-intercept of this line is two. And therefore, the value of 𝑏 must be two. Therefore, I can begin this question by writing down that the equation of this line is 𝑦 equals 𝑚𝑥 plus two. Now I need to find the value of 𝑚, which is the slope of the line. In order to do this, I need two points that lie on the line. I already have one point which has the coordinates zero, two. And if I look carefully at the graph, I can see there’s another point here. I’ve chosen this point because it has integer coordinates. And its coordinates are three, negative two. So I use these two coordinates to calculate the slope of the line. The slope, remember, is found by finding the change in the 𝑦-coordinates divided by the change in 𝑥-coordinates.
Now I could do this formally using 𝑦 one minus 𝑦 two over 𝑥 one minus 𝑥 two. Or I could just look at the graph that we’ve got here. If I look at the triangle that I’ve drawn below the line, then the horizontal part of this triangle gives us the change in 𝑥. I can see then that the change in 𝑥 is three units. Again, looking at the triangle, the vertical part gives me the change in y. And I can see that 𝑦 changes from two to negative two. And therefore, there’s a change in 𝑦 of negative four units. It’s really important that you write that down as negative four, not four, because 𝑦 is decreasing.
So now we have all the relevant information in order to calculate the slope of this line. So the slope of the line is change in 𝑦 over change in 𝑥. It’s negative four over three or negative four-thirds. Finally, I need to substitute this value for the slope into the equation of my line. I have then that 𝑦 is equal to negative four-thirds 𝑥 plus two. I haven’t been asked to do anything else with the equation of this line in terms of rearranging it. So that’s how I’m going to leave my final answer.
In summary then, we’ve seen two different forms of the equation of a line. The slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. And the point-slope form, 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one. We’ve also seen how to find the equation of a line in each of these two forms when we’re given two specific pieces of information, the slope of the line and the coordinates of a point on the line.
