Use the elimination method to solve the simultaneous equations six 𝑥 minus four 𝑦 equals 14, four 𝑥 minus four 𝑦 equals eight.
We’ve been given a pair of linear simultaneous equations in two variables, 𝑥 and 𝑦. And we’re asked to solve these simultaneous equations using the elimination method. This means we need to find the values of 𝑥 and 𝑦 that satisfy both equations.
Now, to use the elimination method means that we need to eliminate or remove one of the variables from our equations. We need to create a new equation in one variable only, either 𝑥 or 𝑦. We do this by considering the coefficients of each of the variables in each of the equations.
If we look carefully at the two equations, we can see that the coefficients of 𝑦 in the two equations are exactly the same. They’re both equal to negative four. This means that we’ll be able to eliminate the variable 𝑦 from this pair of equations. If we subtract one equation from the other, we will eliminate the 𝑦-terms. This is because if we had any number of 𝑦’s and then we take away that exact same number of 𝑦’s, we’ll have no 𝑦’s left. Or more formally, negative four 𝑦 minus negative four 𝑦 is negative four 𝑦 plus four 𝑦, which is zero 𝑦 or simply zero.
We can perform the subtraction in either order. Let’s choose to subtract equation two from equation one, because the coefficient of the other variable 𝑥 is greater in equation one. So if we subtract this way round, we’ll be left with a positive number of 𝑥’s. When we subtract the entirety of equation two from equation one then, we get six 𝑥 minus four 𝑥, which is two 𝑥. Negative four 𝑦 minus negative four 𝑦 is zero, as we’ve already discussed. And on the right-hand side, 14 minus eight is equal to six. So we have the equation two 𝑥 equals six, and we’ve successfully eliminated the 𝑦-variable. We can solve this equation for 𝑥 by dividing both sides by two. And we find that 𝑥 is equal to three.
So we found the value of one of the unknowns, and now we need to find the value of the other. In order to do this, we need to substitute the value of 𝑥 we’ve just found into either of the two equations. In this question, the two equations are equally as complicated or as simple as each other. So let’s choose to substitute into equation two. This gives four multiplied by three minus four 𝑦 is equal to eight. That simplifies to 12 minus four 𝑦 is equal to eight. And then subtracting 12 from each side of the equation, we find that negative four 𝑦 is equal to negative four. Dividing both sides of this equation by negative four, we have 𝑦 equals negative four over negative four, which is equal to one. So we found the values of both variables. 𝑥 is equal to three, and 𝑦 is equal to one.
But we should check our answer. As we substituted our value of 𝑥 into equation two to find the value of 𝑦, we should check the values of both 𝑥 and 𝑦 by seeing if they satisfy the other equation, equation one. Substituting 𝑥 equals three and 𝑦 equals one into the expression on the left-hand side of this equation gives six multiplied by three minus four multiplied by one. That’s 18 minus four, which is equal to 14. As this is the same as the value on the right-hand side of equation one, this confirms that our solution is correct, and so we have our answer.
Now, there is actually a helpful acronym that we can use to help us decide whether we need to add or subtract two equations when we’re using the elimination method. It’s this, SSS, which stands for same signs subtract. This tells us that if the coefficients of a variable are the same magnitude and they have the same sign, then we can eliminate that variable by subtracting one equation from the other.
That’s what we did here, as the coefficients of 𝑦 were both negative four. So they were the same magnitude and they were both negative. They had the same sign. We subtracted one equation from the other in order to eliminate the 𝑦-term. If, however, the coefficients had been of the same size or magnitude but their signs had been different, so one positive and one negative, then we would’ve been able to eliminate the variable by adding the two equations together. That may be intuitive, but this simple acronym can help us remember.
Using the elimination method then, we found that the solution to this pair of simultaneous equations is 𝑥 equals three and 𝑦 equals one.