Testing Solutions for Two-Variable Equations
So given the equation two 𝑥 minus three 𝑦 equals seven, which is the following coordinate pairs is a solution of the equation?
So first of all, with the coordinate pairs we know that the first in the coordinate is always the 𝑥-value and the second is always the 𝑦. So to test if each of these coordinates will be a solution of the equation, what we’ll need to do is substitute the 𝑥-value into the 𝑥 in our equation and the 𝑦-value into the 𝑦 in our equation. And if the left-hand side is equal to the right-hand side, then we know that they are a solution. So what we’re basically trying to do is find which of these coordinates will be on the line — on the line that this equation gives.
So let’s try it out. We’ll have two multiplied by negative two because remember the negative two is the 𝑥-coordinate, and we’re substituting it in for the 𝑥-value then minus three multiplied by two because two was the 𝑦-coordinate, and we’re substituting that in for the 𝑦-value and that should be equal to seven. So two multiplied by negative two is going to be negative four, and then negative three multiplied by two is going to be negative six. Well, negative four minus six is quite clearly not seven; so that is not equal to seven. And as it’s not equal to seven, then this one is not a solution.
So using exactly the same method, let’s try it for the second points. So we’re saying is this coordinate pair a solution of the equation. Well remember that the first is the 𝑥-value and the second is the 𝑦. So we’re going to substitute those in again. So again taking the equation, we’ve got two multiplied by whatever our 𝑥 value is. And in this case, we can see it’s two; so two multiplied by two minus three multiplied by minus one. And what we’re doing here is substituting in if we remember. So if we see the 𝑥, we put what the 𝑥-value is; if we see the 𝑦, we put what the 𝑦-value is. Anyway, so we’re saying that this left-hand side should be equal to seven. So let’s test that. So two multiplied by two is equal to four. And then negative three multiplied by negative one will be positive three, so it’s add three and that is equal to seven. So because the left-hand side is equal to the right-hand side, then we know that this is a solution.
And now let’s try our third and final point. So we should be happy at this point. We can see the three will be what we’ll be substituting for 𝑥 and one will be what we’ll be substituting for 𝑦. So if we write that first line down straight away, so two multiplied by three is six and negative three multiplied by one is negative three. Well six minus three we know is not equal to seven. And as it is not equal to seven, the left doesn’t equal the right; so this is also not a solution. And there we have it. So we can take any coordinate pair and substitute it into any two-variable equation and find out whether it is on the line or curve or whatever equation we’re given. So we could test it with circles or quadratics or any of these things just by substituting in the coordinates.
So let’s try it with one harder example now. So if we’re asked is the point three, twenty-one a solution to the equation 𝑦 equal three 𝑥 squared plus two 𝑥 minus seven, we know that the first point is always the 𝑥-coordinate and the second is always the 𝑦. So we’ll substitute it in. So in this case, we can see that twenty-one as that’s the 𝑦 will be equal to- now be careful with this because of order of operations we’re going to have to put the three inside the brackets again. So I’ll have three multiplied by three squared plus two multiplied by three minus seven.
So let’s try and find out what the right-hand side is actually equal to and whether it is equal to the left-hand side. So we’ve got order of operations; we’ll do three squared first, which gives us nine. So then we’ve got three multiplied by nine which is twenty-seven, and then two multiplied by three is six and minus seven. So then simplifying further, we have twenty-seven add six which is equal to thirty-three, and then taking away seven, we get an answer of twenty-six. So twenty-six is not equal to twenty-one. So that means that the point three, twenty-one is not a solution of this equation. And we have answered the question we need. So all we have to do to test if a point is a solution to a two-variable equation is simply substitute it in and see if one side is equal to the other.