Line one passes through point 𝐴: negative six, 17 and point 𝐵: negative 18, negative 14 and line two passes through the points 𝐶: negative 12, 18 and 𝐷: negative nine, 20. Which of the two lines has a steeper slope?
So we’ve been given the coordinates of two points that lie on each of these lines, and we’ve been asked which line has the steepest slope. We therefore need to calculate the slope of each line and then compare them. First, let’s recall the method for how we calculate the slope of a line. If we have a line that passes through the two points with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two, then the slope of the line is the change in 𝑦 divided by the change in 𝑥. Or more formally, 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one. It doesn’t matter which of the two coordinates we choose to be point one and which we choose to be point two, as long as we’re consistent about the order in which we subtract the 𝑥- and the 𝑦-coordinates.
Let’s look at line one, first of all then. So the slope of the line is 𝑦 two minus 𝑦 one, so that’s negative 14 minus 17. Then we divide it by 𝑥 two minus 𝑥 one, so that’s negative 18 minus negative six. So this simplifies to negative 31 over negative 12, and then the two negatives will cancel each other out in the division. So this leaves us with a slope of 31 over 12 for line one.
Now let’s look at line two. So the slope 𝑦 two minus 𝑦 one, that is 20 minus 18. Now we’re going to divide by 𝑥 two minus 𝑥 one, so that is negative nine minus negative 12. So this simplifies to two-thirds. So we have the slope of both of our lines, line one has a slope of 31 over 12 and line two has a slope of two-thirds.
So we have the slopes of the two lines and we want to compare them to see which is steeper. Now there are a couple of different ways that we could do this. You may be able to see just by looking that line one has the steepest slope because 31 over 12 is an improper fraction, which means it takes a value greater than one, whereas two-thirds is less than one. We could do it more formally by writing these two fractions with the same denominator and then comparing them. If I look at line two and I wanted to have a denominator of 12, then I need to multiply three by four in order to get to 12. If I do the same to the numerator, so I multiply two by four, then I have an equivalent fraction of eight over 12.
So now I’m comparing eight over 12 with 31 over 12. And as I have the same denominator, it’s far more straightforward to see which is greater. So clearly 31 over 12 is greater than eight over 12. And therefore, the answer to the problem which of the two lines has a steeper slope, well it’s line one.
So within this question, we had to recall the method for calculating the slope of a line given the coordinates of two points that lie on the line. We calculated the slope of both, line one and line two. And then formally, compared the two fractions by finding a common denominator.