In this video, we’re going to look at how to find the equation of a straight line in various different forms, given two pieces of information, the slope of the line and the 𝑦-intercept. Firstly, a reminder of the different formats that you may be asked to give the equation of a line in. The first is slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. The letters 𝑚 and 𝑐 each represent particular properties of the straight line. 𝑚 represents the slope of the line, which means, for every one unit you move to the right, the line moves this many units either up or down, depending on whether 𝑚 is positive or negative.
𝑐 represents the 𝑦-intercept of the line, which is the value at which the line cuts the 𝑦-axis. The second common format for the equation of a straight line is point-slope form, 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one. 𝑚 represents the slope of the line, as we saw previously. 𝑥 one, 𝑦 one represents the coordinates of any particular point that lies on this given straight line. Now, in this video, we’re looking at finding the equation of a line given its slope and its 𝑦-intercept. And therefore, it’s usual that we would be using the slope-intercept form in order to do this.
Determine, in slope-intercept form, the equation of the line which has a slope of eight and a 𝑦-intercept of negative four.
So we’re told the way that our answer should be expressed, slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. So what we need to do is determine the values of 𝑚 and 𝑐 for this question. We’re actually given the values of 𝑚 and 𝑐 explicitly within the information in the question itself. We’re told that this line has a slope of eight; this means that the value of 𝑚 is eight. We’re also told that the line has a 𝑦-intercept of negative four; this means that the value of 𝑐 is negative four. So all we need to do to answer this question is substitute the values of eight and negative four into the slope-intercept form of a straight line. Therefore, we have that the equation of the straight line is 𝑦 equals eight 𝑥 minus four.
Find the coordinates of the point where 𝑦 equals four 𝑥 plus 12 intersects the 𝑦-axis.
So in this question, we’re given the equation of a straight line in slope-intercept form. And we’re asked for the coordinates of the point where it intersects the 𝑦-axis. What this question is checking then is do we understand slope-intercept form and what the different parts of the equation represent. Remember that slope-intercept form is 𝑦 equals 𝑚𝑥 plus 𝑐, where 𝑐 represents the 𝑦-intercept of the straight line, which is what we’re looking for here. The 𝑦-intercept is the point where the line intersects the 𝑦-axis. So we can see by comparing the general form and the specific straight line that we have, the value of 𝑐 here is 12.
But the question doesn’t just ask us for the value of 𝑐; it asks us for the coordinates of this point. So the 𝑦-intercept remember is a point on the 𝑦-axis. We’ve just worked out its 𝑦-coordinate; it’s this value of 12. To work out the 𝑥-coordinate, we just need to remember that at every point on the 𝑦-axis the 𝑥-coordinate is zero. You could perhaps see this more clearly by picturing what the graph would look like as I’ve done here. So the coordinates of this point then are going to be zero, 12. And that is our final answer to this question.
Write the equation represented by the graph shown. Give your answer in the form 𝑦 equals 𝑚𝑥 plus 𝑏.
So we have a diagram of a straight line. And we’re asked to give its equation in slope-intercept form, which means we need to work out what each of these two things are. From looking at the diagram, we can see that the 𝑦-intercept is negative four, which means that the value of 𝑏, which is the letter used here to represent the 𝑦-intercept, must be negative four. So I can write down the beginnings of the equation of this straight line; it’s 𝑦 equals 𝑚𝑥 minus four. Next, we need to find the value of 𝑚, the slope of this line. And in order to do this, I need the coordinates of two points that lie on the line.
We’ve already had identified one point, the point with coordinates zero, negative four. Looking at the graph, I can also see that there’s a point here that would be convenient to use. This point lies on the 𝑥-axis and has the coordinates six, zero. So I’m going to use these two points to calculate the slope of the line. So, the slope of the line can be calculated as a change in 𝑦 divided by a change in 𝑥. Or you can think of this as 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one, if you choose to label the two points as 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. I’m just going to look at the diagram in order to work out the change in 𝑦 and the change in 𝑥.
The change in 𝑦 first of all then, well that is the vertical length in this triangle. And I can see that it moves from a 𝑦-coordinate of negative four to a 𝑦-coordinate of zero. Therefore, the change in 𝑦 is positive four. Now, let’s look at the change in 𝑥; this is the horizontal change. So I can see from the diagram that this moves from a value of zero to a value of six, which gives me a change in 𝑥 of positive six. So the slope of this line then is four over six. But this can be written as a simplified fraction; it’s two-thirds. Finally then, I just need to substitute this value of 𝑚, the slope of the line, into the equation. So the equation of the line represented by this graph is 𝑦 equals two-thirds 𝑥 minus four.
Find the equation of the line with 𝑥-intercept three and 𝑦-intercept seven and calculate the area of the triangle on this line and the two coordinate axes.
So this question has two parts. We’re first asked to find the equation of a line and then we’re asked to calculate the area of this triangle. I think a diagram would be helpful here in order to visualize the situation. So we have a pair of coordinate axes. We’re told that this line has 𝑥-intercept three, which means it cuts the 𝑥-axis at three. We’re also told the line has 𝑦-intercept seven, so it cuts the 𝑦-axis at seven. By connecting these two points, I have the line that I’m looking to find the equation of and I can see the triangle that I’m asked to find the area of. It’s this triangle here.
So let’s start with the first part of this question, which asks to find the equation of this line. I’m going to do this using the slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐. I can work out one of these two values straight away. Remember 𝑐 represents the 𝑦-intercept of the line. And I’m told in the question that this is equal to seven. So the equation of the line is 𝑦 equals 𝑚𝑥 plus seven. I now need to work out the slope of this line. And in order to do so, I need the coordinates of two points on the line. Well, I can use the coordinates of these points, the 𝑥-intercept and the 𝑦-intercept.
The slope of the line remember is calculated as the change in 𝑦 divided by the change in 𝑥. So looking at my diagram and using these two points, I’m gonna find the change in 𝑦 first of all. I can see that as I move from left to right across the diagram, the 𝑦-coordinate changes from seven to zero, which is a change of negative seven. It’s really important that you consider this change in 𝑦 as negative seven, not seven. The line is sloping downwards from left to right, and therefore it has a negative gradient. Now, let’s look at the change in 𝑥. I can see that as I move from left to right across this diagram, the 𝑥-coordinate changes from zero to three, which gives me a change in 𝑥 of positive three.
Now, I can substitute the change in 𝑦 and the change in 𝑥 into my calculation for the slope of this line. And we have that the slope of the line is equal to negative seven over three. Finally, in order to complete the first part of the question and find the equation of the line, I need to substitute this value for 𝑚 into the equation. I have then that the equation of this line is 𝑦 equals negative seven over three 𝑥 plus seven. Now, sometimes you may be asked to give your answer in a slightly different format, for example, a format that doesn’t involve fractions. So you’d need to multiply the equation there by three, but as it hasn’t been specified here I’m going to leave my answer as it is now. So that’s the first part of the question completed.
The second part asked me to calculate the area of the triangle formed by this line and the two coordinate axes. Now from the diagram, we can see that this is a right-angled triangle because the 𝑥- and 𝑦-axes meet at a right angle. To find the area of a right-angled triangle, we need to multiply the base by the perpendicular height and then divide by two. So looking at the diagram, I can see that the base of this triangle is that measurement of three units. The height of the triangle is seven units. Now, we refer to this as negative seven when we’re calculating the slope of the line because the direction was important. For when we’re just looking at the length of that line in order to calculate an area, we’ll take its positive value of seven. So our calculation for the area is three multiplied by seven divided by two. And this gives us an answer of 10.5 square units for the area of this triangle.
In summary then, when we’re given the slope and the 𝑦-intercept of a straight line, it’s usual to calculate its equation in slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐 because the values can be just substituted directly into this form. We may need to calculate the slope of the line ourselves given two points on the line by using change in 𝑦 divided by change in 𝑥. It would also be possible to give the equation of a line in point-slope form, but this would be unnecessarily complicated if the information we’re given is slope and 𝑦-intercept as this aligns so conveniently with the slope-intercept form.