### Video Transcript

Find the slopes of line π΄π΅, line π΅πΆ, and line πΆπ·.

In this question, weβre asked to find the slopes of the three different line sections in our graph. The slope of a line is the amount that the π¦-value increases or decreases as we increase the π₯-value. And we can calculate it by finding the change in π¦-values divided by the change in π₯-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 π¦-values are decreasing.

So letβs start by finding the slope of line π΄π΅. So letβs begin by picking two points on the line π΄π΅. 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 π₯-value comes before the π¦-value. So to calculate the slope of π΄π΅, we write 50 minus 10 since we take the π¦-value in our second coordinate, 50. And we subtract the π¦-value in our first coordinate, 10.

To work out the change in π₯, we take the π₯-value in our second coordinate, four, and subtract the π₯-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 π΄π΅ will be 10. But letβs see what would happen if we had picked different coordinates on our line π΄π΅. Letβs say weβd picked the coordinates one, 20 and three, 40. Would that have given us a different slope?

Well, in this case, when weβre working out the change in π¦, the values that we would get would be 40 minus 20. And our change in π₯-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 doesnβt matter which coordinates we use for the slope. Any pair of coordinates will give us the same value for the slope.

Letβs now calculate the slope of line π΅πΆ. Looking at our graph, we can see that line π΅πΆ is a horizontal line. So letβs 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 π¦-value calculation as 50 minus 50. And the change in π₯-values will be seven minus four, which we can simplify to zero over three, which means that the slope of line π΅πΆ will be equal to zero. If we recall that the slope of the line is the amount that the π¦-value increases or decreases as we increase π₯, in this case, the π¦-value didnβt increase at all when we increased the π₯-value.

So letβs move on then to finding the slope of line πΆπ·. 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 π¦-value, we need to be a bit careful. It can be tempting to write 50 minus 40. But since we always do the second π¦-value, subtract the first π¦-value, this would give us 40 minus 50. Our change in π₯-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 π¦-value is decreasing when π₯ increases. And since negative 10 divided by one is the same as negative 10, this means that the slope of line πΆπ· is negative 10. And any line sloping downwards, like line πΆπ·, will always have a negative value for slope.

And so our final answer is the slope of line π΄π΅ is 10. The slope of line π΅πΆ is zero. And the slope of line πΆπ· is negative 10.