Find the slopes of line 𝐴𝐵, line 𝐵𝐶, and line 𝐶𝐷.
In this question, we’re asked to find the slopes of the three different line sections in our graph. The slope of a line is the amount that the 𝑦-value increases or decreases as we increase the 𝑥-value. And we can calculate it by finding the change in 𝑦-values divided by the change in 𝑥-values. The slope of a line is also often referred to as the rise over the run. And there will be times when the rise has a negative value if the 𝑦-values are decreasing.
So let’s start by finding the slope of line 𝐴𝐵. So let’s begin by picking two points on the line 𝐴𝐵. We can pick any points. But often the best ones to pick are those which clearly lie on the lines on our coordinate grid. We can see here that the coordinate zero, 10 is clearly on the line as is the coordinate four, 50. In each of our coordinates, the 𝑥-value comes before the 𝑦-value. So to calculate the slope of 𝐴𝐵, we write 50 minus 10 since we take the 𝑦-value in our second coordinate, 50. And we subtract the 𝑦-value in our first coordinate, 10.
To work out the change in 𝑥, we take the 𝑥-value in our second coordinate, four, and subtract the 𝑥-value in our first coordinate, zero, giving us four minus zero. We can then simplify this, giving us 40 over four. And then, since 40 over four is the same as 40 divided by four, this means that the slope of line 𝐴𝐵 will be 10. But let’s see what would happen if we had picked different coordinates on our line 𝐴𝐵. Let’s say we’d picked the coordinates one, 20 and three, 40. Would that have given us a different slope?
Well, in this case, when we’re working out the change in 𝑦, the values that we would get would be 40 minus 20. And our change in 𝑥-values on the denominator would be three minus one. Simplifying that would give us 20 over two, which would give us the same value 10 for our slope. And so it doesn’t matter which coordinates we use for the slope. Any pair of coordinates will give us the same value for the slope.
Let’s now calculate the slope of line 𝐵𝐶. Looking at our graph, we can see that line 𝐵𝐶 is a horizontal line. So let’s see how we can calculate a numerical value for the slope. We can use the coordinate four, 50. And we can also see that the coordinate seven, 50 will lie on the line. So this will give us a change in 𝑦-value calculation as 50 minus 50. And the change in 𝑥-values will be seven minus four, which we can simplify to zero over three, which means that the slope of line 𝐵𝐶 will be equal to zero. If we recall that the slope of the line is the amount that the 𝑦-value increases or decreases as we increase 𝑥, in this case, the 𝑦-value didn’t increase at all when we increased the 𝑥-value.
So let’s move on then to finding the slope of line 𝐶𝐷. And we can use the coordinates seven, 50 and eight, 40 which lie on this line. When it comes to working out the change in 𝑦-value, we need to be a bit careful. It can be tempting to write 50 minus 40. But since we always do the second 𝑦-value, subtract the first 𝑦-value, this would give us 40 minus 50. Our change in 𝑥-values will be eight minus seven. We can simplify this, giving us negative 10 over one. So in this case, the rise of our graph is a negative value as the 𝑦-value is decreasing when 𝑥 increases. And since negative 10 divided by one is the same as negative 10, this means that the slope of line 𝐶𝐷 is negative 10. And any line sloping downwards, like line 𝐶𝐷, will always have a negative value for slope.
And so our final answer is the slope of line 𝐴𝐵 is 10. The slope of line 𝐵𝐶 is zero. And the slope of line 𝐶𝐷 is negative 10.