In the given diagram, 𝐴𝐵 is a straight line. Answer the following questions. Form an equation that will allow you to calculate 𝑥. And then, find the value of 𝑥.
Let’s take a moment to look at the diagram that we’ve been given as part of this problem. Firstly, we can see a line with two points on it marked 𝐴 and 𝐵. And we’re told, in the given diagram, 𝐴𝐵 is a straight line. Now, this is important. What do we know about angles on a straight line? Well, firstly, we know that a straight-line angle is worth 180 degrees. And so any two angles that are on a straight line, no matter how they’re drawn, will have a total of 180 degrees. And we can use these facts about straight-line angles to help us to solve this particular problem.
In the diagram, we can see that our 180-degree angle has been split into two separate angles. But interestingly, both angles aren’t labelled with a number. We have an expression to tell us how many degrees they’re worth. The angle on the left is labelled as two 𝑥 plus 10 degrees. And the angle on the right is labelled three 𝑥 plus 10 degrees. The first part of our problem asks us to form an equation that will allow us to calculate 𝑥. Actually, calculating 𝑥 comes in the second part of the problem. So in this first part, we just need to make an equation that will help us to do so.
Well, what can we say about our two angles? Well, we can apply our knowledge of straight-line angles here. And we can say that if we add the two expressions together, we’re going to get the number 180. So in other words, two 𝑥 plus 10, that’s the angle on the left, plus three 𝑥 plus 10, so we’ve added the angle on the right, equals 180. And if we look at the expression that we’ve made, we can see it’s just one long addition. And we have some terms here that are the same. And we can group them together. We have some amount of 𝑥. And also, we have some numbers that we’re adding. So if we group together these like terms, we can write the equation more simply.
So let’s begin by looking at how many lots of 𝑥 we have. Two 𝑥 means two lots of 𝑥. And, of course, three 𝑥 means three lots of 𝑥. So all together, we’re being asked to calculate five lots of 𝑥. That’s a much simpler way of writing the two terms. And we’re also being asked to add some numbers. We have 10 and 10. This makes a total of 20. And so we can write the equation much more simply as five 𝑥 plus 20 equals 180.
Now, we can use this equation to find the value of 𝑥 as it asks us to do in part two. We can think of an equation like this as being like a set of balances. The equal sign tells us that both sides of our set of balances are worth the same. So on one side, we have five 𝑥 plus 20. And that’s worth exactly the same as 180. And we want to end up with just the letter 𝑥 on the left-hand side and an expression that says 𝑥 equals something.
So let’s start getting rid of these terms on the left-hand side. The first thing we can do is get rid of that 20. If we take away 20 from five 𝑥 plus 20, we’re left with just five 𝑥. But remember, just like a set of balances, both sides of our equation need to be worth the same. So if we subtract 20 from one side, we need to subtract 20 from the other. 180 take away 20 equals 160. We’re now left with the equation five 𝑥 equals 160 or five lots of 𝑥 equals 160. To find out what one lot of 𝑥 is worth, we need to divide both sides of our equation by five. So 𝑥 is going to be worth 160 divided by five. How many fives are in 160?
One way to find the answer might be to count in groups of 10 lots of five or 50s. We know 50 is 10 lots of five. So 100 must be 20 lots of five. Another lot of 50, 150, would be 30 lots of five. And 160 is another 10. So that’s two more lots of five, 32 lots of five. And so we can say 160 divided by five equals 32. You know the interesting thing about this problem? Nowhere does it ask us to calculate what each angle is worth in the diagram.
Now that we know what 𝑥 is worth, of course, we could calculate the value of each angle. But that’s not what the questions asked us to do. Instead, we needed to apply our knowledge of straight-line angles and the fact that they always equal 180 degrees to form an equation that allowed us to calculate 𝑥. We wrote an equation. And then, we simplified it to get the equation five 𝑥 plus 20 equals 180. And we then used this equation to help us calculate the value of 𝑥, which we found was 32.