Solve the simultaneous equations 𝑥 plus four 𝑦 is equal to 17 and two 𝑥 plus seven 𝑦 is equal to five.
Well, these are simultaneous equations so I’m going to put these curly braces here to indicate that they’re both true at the same time, and I’m also going to number them. This is equation one, and this is equation two.
Now the numbering means that I can refer to each equation individually as I’m going through and annotating my working out. And looking at both of these equations, I’ve got 𝑥 in the first equation and two 𝑥 in the second equation; I’ve got four 𝑦 in the first and seven 𝑦 in the second. I can’t simply add or subtract one of those equations from the other to eliminate one of the variables, so what I need to do is to multiply or divide through in one of the equations to get the same number of 𝑥s or 𝑦s in both equations.
Now the easiest way to do this is to double everything in equation one. So doubling everything in equation one, I’ve got two lots of 𝑥 are two 𝑥, and two lots of positive four 𝑦 are positive eight 𝑦, and two lots of 17 are 34, and we can call this equation number three. Now if I look at equations two and three, I’ve got two 𝑥 in both of them, and I’ve got seven 𝑦 in one, eight 𝑦 in the other, and then equals five and equals 34.
So if I take one equation away from the other, I’m going to eliminate the 𝑥 variable. But which one should I take away from which? Well if I do equation two take away equation three, two 𝑥 minus two 𝑥 is nothing, seven 𝑦 take away eight 𝑦 is gonna leave me with negative 𝑦. But if I did equation three take away equation two, eight 𝑦 minus seven 𝑦 will be a positive one 𝑦, and two 𝑥 minus two 𝑥 is still zero. So that’s what I’m gonna do.
So equation three take away equation two, two 𝑥 take away two 𝑥 is nothing; eight 𝑦 take away seven 𝑦 is just 𝑦; and 34 take away five is 29. So 𝑦 is equal to 29. Now I can use that value for 𝑦 in one of my earlier equations — one, two, or three — to find out the corresponding value of 𝑥. Now the easiest equation to do that with is equation one here.
So substituting 𝑦 equals 29 into equation one, I’ve got 𝑥 plus four times 29, four times 𝑦, is equal to 17; and four times 29 is 116. So 𝑥 plus 116 is equal to 17. Now if I take away 116 from both sides, I’m just gonna leave myself with 𝑥 on the left-hand side, and 17 take-away 116 is negative 99. So now I’ve got a value for 𝑥. Now I can check my answer by substituting in those values I’ve got for 𝑥 and 𝑦 into one of my equations.
Well I’ve already used equation one to work out the value of 𝑥, so I’m gonna use equation two to do my check. So substituting in, 𝑥 equals negative 99, and 𝑦 equals 29. I’ve got two times negative 99 plus seven times 29 is equal to five. Well two times negative 99 is negative 198, and seven times 29 is 203. So negative 198 plus 203 is equal to five? Well, yes it is!
So that means that the values I used for 𝑥 and 𝑦 must be correct. So I can write my answer out nice and clearly: 𝑥 equals negative 99, and 𝑦 equals 29. And I know that my answer is correct because I’ve checked it.