Answer the following questions using the given diagram. Firstly, form an equation that will allow you to calculate 𝑥. Secondly, find the value of 𝑥.
So within this question, we’ve been given a diagram. And by looking at the diagram, we can see that two of the angles have been labelled in terms of this variable 𝑥. We need to use some angle facts in order to first form an equation and then solve our equation in order to find the value of 𝑥.
The diagram consists of a pair of straight lines, which intersect. And by looking at the diagram, we can see that the two angles that have been labelled are a particular type of angles that was referred to as vertically opposite angles. These are the pair of angles opposite one another when a pair of straight lines intersect.
There is also another pair of vertically opposite angles in the diagram, the pair of angles that I’ve marked in green. However, we’re interested in the pair of angles marked in blue, and also now in orange.
In order to answer this question, we need to remember a key fact about vertically opposite angles, which is that vertically opposite angles are equal to each other. So this gives us an idea for how to answer the first part of the question: form an equation that will allow you to calculate 𝑥.
Well, if vertically opposite angles are equal to each other, then these two expressions for the angles must also be equal to each other. So we have that 15𝑥 plus 50 is equal to five 𝑥 plus 60. And there is our unsimplified equation that we can use in order to calculate 𝑥.
So we’ve answered the first part of the question, and now we need to go on and answer the second part of the question, which asked us to find the value of 𝑥, or, in other words, solve this equation. So the problem is now an algebraic one, where we’re looking to solve this linear equation.
So looking at the equation, I can see that there are currently 𝑥 terms on both sides of the equation, and I’d like to collect all the 𝑥 terms on the same side, the left-hand side. So my first step towards solving this equation is just subtract five 𝑥 from both sides. So when I subtract five 𝑥 from the left-hand side, I’m left with 10𝑥 plus 50, and when I subtract five 𝑥 from the right-hand side, I’m just left with 60.
Next, I want to have the 𝑥 terms on their own, so I need to subtract 50 from both sides of the equation. And having done so, I now have that 10𝑥 is equal to 10. The final step in order to find the value of 𝑥 is that I need to divide both sides of this equation by 10, as I want to go from 10𝑥 to one 𝑥. So dividing both sides of the equation by 10 gives me that 𝑥 is equal to one.
So to answer this question, we had to recall a key angles fact, which is that vertically opposite angles are equal to one another. Once we’d recalled this fact, we were able to set up our equation to answer the first part of the question: 15𝑥 plus 50 is equal to five 𝑥 plus 60. We were then able to solve this linear equation using our algebraic methods to give us the value of 𝑥: 𝑥 is equal to one.