The graph shows the capital of a company over eight years. Determine the slope of the line which represents the increase in capital in the first three years.
This is a really nice question because it actually starts to talk about slope in a real-life context, which is good because it actually gets us to think about how it could be useful. The reason that slope can be useful when we’re looking at this real-life context, is that actually we waana see how quickly the capital increases or decreases over a set time. And for this question, it’s the first three years. As we’re looking at the first three years, it’s this section in the graph that we’re actually going to concentrate on when trying to solve this problem.
So what does slope actually mean? Well, slope or 𝑚 is the change in 𝑦 divided by the change in 𝑥. So in this context, it’s gonna be our change in capital divided by our change in the number of years. To help us calculate slope, we actually have an equation which is 𝑚 is equal to 𝑦 two minus 𝑦 one divided by 𝑥 two minus 𝑥 one where 𝑦 one, 𝑦 two and 𝑥 one, 𝑥 two are actually the coordinates of two points on our line. When calculating the slope, you need to actually pick two points on your line cause actually if it’s a straight line, your slope will be the same throughout the whole length of that line. So it doesn’t really matter where you pick them, but do make sure they’re in a place where there is obvious values.
So here we’ve already got 𝐴 and 𝐵 marked on our diagram. So we’ve got 𝐴 which is naught, 60 and 𝐵 which is three, 70. And now, I’m gonna label our coordinates. And I probably recommend you do this just until you’re really confident using the formula. Now we’ve labelled our coordinates, we can actually substitute our values into our formula. So first of all, we have 70 which is 𝑦 two minus 60 which is 𝑦 one divided by three which is 𝑥 two minus zero which is 𝑥 one. Great! So we’ve now subbed all of our values from our coordinates into our slope formula, and we can now solve this to find our slope.
Fab! So we’ve now found the slope of our line, 𝑚 is equal to 10 over three or ten-thirds. And so therefore, we’ve solved the problem because what we’ve actually done, is we found the slope which represents the increase in capital in the first three years. And what that actually means in context, is actually over the first three years, our capital of the company has increased by 10000 pounds.