Video Transcript
The system of equations a third times eight 𝑥 minus two 𝑦 equals seventeen thirds and 𝑦 equals three 𝑥 has the solution 𝑥, 𝑦. What is the value of 𝑥?
So the question has told us that we’ve got one solution. And when we have a system of equations, it’s a good idea to write them out like this. Let’s call the first equation, equation number one and the second equation, equation number two. Now, if you had just one of these equations, we wouldn’t be able to solve uniquely for 𝑥 and 𝑦, because we’d have one equation with two different variables. But in equation number two, we can see that 𝑦 is equal to three times the value of 𝑥. And we can use this information in equation one. Wherever we see 𝑦 in the first equation, we can replace it with three 𝑥. And when we do that, we’ve got a third times eight 𝑥 minus two times three 𝑥 is equal to seventeen thirds. Now, two times three 𝑥 is six 𝑥. So we can simplify that equation to a third times eight 𝑥 minus six 𝑥 is equal to seventeen thirds. And eight 𝑥 minus six 𝑥 is two 𝑥. So we can simplify even further. A third times two 𝑥 is equal to seventeen thirds.
Now, we’re trying to solve this equation to find the value of 𝑥. So if I multiply both sides of the equation by three or, more helpfully, three over one, then the numerator and the denominator on the left-hand side are both divisible by three. Three divided by three is one. And three divided by three is one. So I’ve just got two 𝑥 on the left-hand side. And likewise, on the right-hand side, the numerator is divisible by three. And the denominator is divisible by three. So I’m just left with 17. So two 𝑥 is equal to 17. If I divide both sides of my equation by two now, I can divide top and bottom by two on the left-hand side to just leave me with 𝑥. And I’ve got 17 over two on the right-hand side. So the answer is 𝑥 equals 17 over two. Or, if you prefer to express your number as a decimal, 𝑥 is equal to 8.5.
Now, the question only asked us to find the value of 𝑥. But if it had asked us to find the value of 𝑦, that would be quite easy. We could just substitute the value we found for 𝑥 into equation two. And we find that 𝑦 is equal to three times 17 over two. And three times 17 is 51. So 𝑦 is equal to 51 over two or as a decimal 25.5. But coming back to the question we were actually asked, the value of 𝑥 is 17 over two or 8.5.
