In this video, we’re going to see how to find the equation of a straight line in various different forms given the coordinates of two points that lie on the line. First, a reminder of two the different formats for the equation of a straight line.
The first of these is slope-intercept form: 𝑦 equals 𝑚𝑥 plus 𝑐. Here, 𝑚 and 𝑐 have specific meanings related to properties of the line. 𝑚 represents the slope or steepness of the line. As you move along line from left to right, for every one unit that you travel horizontally, you move 𝑚 units either up or down depending on where 𝑚 is positive or negative. 𝑐 represents the 𝑦-intercept, which is the 𝑦-value at which the line intersects the 𝑦-axis.
The second commonly used form for the equation of a straight line is point-slope form 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one. Here, 𝑚 represents the slope of the line as we just discussed. 𝑥 one, 𝑦 one represents the coordinates of any point on the line, so not just specifically the 𝑦-intercept but any point along the length of the line. In this video, we’ll see how to find the equation of a straight line in each of these two forms when we’ve been given the coordinates of two points that lie on the line.
Find the equation of the straight line which passes through the two points negative one, negative two and negative seven, negative four. So within this question, we’re given the coordinates of two points that lie on the line, and we were asked to find its equation. We haven’t been asked an equation of particular format, so I’m going to choose to use the point-slope method, 𝑦 minus 𝑦 one equals 𝑚 𝑥 minus 𝑥 one.
First of all, I want to calculate the slope of this line, the value of 𝑚. The slope can be calculated using change in 𝑦 divided by change in 𝑥, or more formally 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one where 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two are the coordinates of the two points on the line. It doesn’t matter which point I choose to be 𝑥 one, 𝑦 one and which I choose to be 𝑥 two, 𝑦 two, so I’ve chosen to do it this way round.
Now I need to substitute the values into the calculation for the slope. First of all, the change in 𝑦 is negative four minus negative two. The change in 𝑥 is negative seven minus negative one. This is equal to negative two divided by negative six, which simplifies to one-third as the two negatives cancel each other out. So we know the slope of the line is one-third. Now I can substitute this into my equation for the line; I then have 𝑦 minus 𝑦 one equals one-third 𝑥 minus 𝑥 one.
Next, I need to substitute the values of 𝑥 one and 𝑦 one into the equation of the line. Now it doesn’t have to be the values of negative one and negative two; it could equally be the values of negative seven and negative four that I use, whichever point I substitute will give the same result for the overall equation of the line. I am going to choose to use the point with coordinates negative one and negative two. So I substitute negative two for 𝑦 one and negative one for 𝑥 one. And now I have 𝑦 minus negative two is equal to one-third 𝑥 minus negative one. The double negatives simplify to give 𝑦 plus two is equal to one-third of 𝑥 plus one.
Expanding the bracket on the right side of the equation gives 𝑦 plus two equals one-third 𝑥 plus one-third. Finally, I’m going to subtract two from both sides of this equation. This gives me 𝑦 equals one-third 𝑥 plus one-third minus two. Now two can be written as six over three, so what I have is 𝑦 equals one-third 𝑥 plus one-third minus six-thirds. So this simplifies to give my final answer to the question, which is 𝑦 is equal to one-third 𝑥 minus five over three.
Now as we haven’t been asked to give her answer in a particular format, that’s perfectly acceptable to have some fractions involved. Alternatively, you may choose to eliminate the fractions by multiplying the equation through by three. So one alternative way to leave your answer for this problem would be as three 𝑦 is equal to 𝑥 minus five.
Find the equation of the straight line represented by the graph below in the form of 𝑦 equals 𝑚𝑥 plus 𝑐. So before we begin, a couple of observations. We’ve been asked to give our answer in the form 𝑦 equals 𝑚𝑥 plus 𝑐, which is slope-intercept form. Looking at the graph, we can see that the line cuts the 𝑦-axis somewhere between one and two, but we can’t see the exact value of the 𝑦-intercept from the diagram. We’re therefore going to need to calculate it rather than reading its value directly from the graph.
Looking at the graph more closely, we can see that there are two points they have been highlighted with blue dots. These points are each placed directly on the intersection of some of the gridlines, and therefore they both have integer coordinates. We have the points negative two, seven and one, negative one. So we’re going to use these two points in order to find the equation of the line. We’ll start off by calculating the slope of the line.
The slope of line can be found using change in 𝑦 divided by change in 𝑥. By drawing in this right-angled triangle below the line, we can see the change in 𝑦 and the change in 𝑥 more easily. Let’s look at the change in 𝑦 first of all. As we move across the graph from left to right, we can see that the 𝑦-value changes from seven to negative one. This is a change of negative eight. It’s very important that we count this change as negative eight, not eight; the 𝑦-value is decreasing, and therefore it’s a negative value.
Now let’s look at the change in 𝑥. We can see that as we move across the graph from left to right, the 𝑥 value changes from negative two to one, which is a change of positive three units. So now we can substitute the values into our calculation of the slope. And we have that 𝑚 is equal to negative eight over three. The next step is to substitute this value of 𝑚 into the slope-intercept form of the line: 𝑦 is equal to negative eight over three 𝑥 plus 𝑐. Now we need to calculate the value of 𝑐, the 𝑦-intercept.
And as we’ve already said it isn’t easy to identify this exactly from the graph, so we need an alternative approach. What we’re going to do is take one of these coordinates, and I’m going to choose to use the point with coordinates negative two, seven. This point lies on the line so it tells me that within this equation, when 𝑥 is equal to
two [negative two], 𝑦 must be equal to seven. So I can substitute the values of 𝑥 and 𝑦 into this equation in order to find 𝑐. So substituting negative two for 𝑥 and seven for 𝑦, we have seven is equal to negative eight over three multiplied by negative two plus 𝑐. This simplifies to seven is equal to 16 over three plus 𝑐.
I want to solve this equation to find the value of 𝑐, so I need to subtract 16 over three from both sides. At the same time, I’ll express seven as 21 over three. So now I have 𝑐 is equal to 21 over three minus 16 over three, which gives me a value of five over three. The final step is I need to substitute this value of 𝑐 into the equation of the line. The equation of the line then in the requested form is 𝑦 equals negative eight over three 𝑥 plus five over three.
Find the linear equation describing the relation in the table, then find the value of 𝐴. So we have here a table which has values of 𝑥 and values of 𝑓 of 𝑥, and we’re told that they are related by our linear equation. This means that the relationship between 𝑥 and 𝑓 of 𝑥 will be of the form 𝑓 of 𝑥 equals 𝑎𝑥 plus 𝑏 where 𝑎 and 𝑏 represent the values of constants. This is the same structure as the equation of a straight line, 𝑦 equals 𝑚𝑥 plus 𝑐. But instead of using 𝑚 and 𝑐, we’re using the letters 𝑎 and 𝑏 to represent the constants.
So we need to find the values of 𝑎 and 𝑏. And to do this we can look at the pairs of 𝑥 and 𝑓 of 𝑥 that we already know. First, we know that when 𝑥 is equal to negative five, 𝑓 of 𝑥 is equal to negative two. We can substitute these values into the linear equation to give us our first relation. We have negative two is equal to negative five 𝑎 plus 𝑏. We also know that when 𝑥 is equal to negative eight, 𝑓 of 𝑥 is equal to negative seven, and so we can also substitute this pair of values into the relation. We have negative seven is equal to negative eight 𝑎 plus 𝑏. Now what we have are a pair of simultaneous equations which we can solve in order to find the values of 𝑎 and 𝑏. If we were to subtract equation two from equation one, this would eliminate the 𝑏 terms. We’d be left with with negative two minus negative seven is equal to negative five 𝑎 minus negative eight 𝑎. This simplifies to five is equal to three 𝑎. Now we can solve this to find the value of 𝑎 by dividing both sides of the equation by three. This gives us 𝑎 is equal to five over three. Now note this is lowercase 𝑎, one of the letters describing the relation, not the capital 𝐴 which we need to find later in the question.
So substituting this value of 𝑎 back into the relation, we now have that 𝑓 of 𝑥 is equal to five over three 𝑥 plus 𝑏. We found one of these two values. We now need to find the value of 𝑏. Let’s use that first known pair of values for 𝑥 and 𝑓 of 𝑥. Substituting 𝑥 equals negative five and 𝑓 of 𝑥 equals negative two, we now have the equation negative two equals five over three multiplied by negative five plus 𝑏.
And this equation can be solved to find the value of 𝑏. We have negative two is equal to negative 25 over three plus 𝑏. And an order to find 𝑏, I need to add 25 over three to both sides of the equation. At the same time, I’m going to express negative two as negative six over three so that I can add these two terms together more easily. I have 𝑏 is equal to negative six over three plus 25 over three, which simplifies to 19 over three.
So now I have the values of 𝑎 and 𝑏, so I can substitute them both into the linear equation to describe this relation. So we have 𝑓 of 𝑥 is equal to five over three 𝑥 plus 19 over three. Now I haven’t quite finished because the question also asked me to find the value of 𝐴, capital 𝐴, which is the value 𝑓 of 𝑥 takes when 𝑥 is equal to one.
In order to find this, I need to substitute 𝑥 equals one into the relation we’ve just found. So this tells me that 𝐴 is equal to five over three multiplied by one plus 19 over three. This simplifies to 24 over three, and 24 over three is just eight. So our answer to the second part of the question is that the value of 𝐴 is eight.
In summary then, we’ve seen how to find the equation of a straight line in either of these two forms: slope-intercept form, 𝑦 equals 𝑚𝑥 plus 𝑐, and point-slope form, 𝑦 minus 𝑦 one = 𝑚 𝑥 minus 𝑥 one. When we’re given the coordinates of two points on the line, we first have to calculate the slope of the line using change in 𝑦 over change in 𝑥 or 𝑦 two minus 𝑦 one over 𝑥 two minus 𝑥 one. We can then use either the slope-intercept method or the point-slope method in order to calculate the 𝑦-intercept of the line and hence find its equation.