Two function machines are shown. A: input multiplied by six subtract two output. B: input multiplied by two add eight output. Both function machines have the same input. Work out the value of the input such that the output of A is two times the output of B.
We’re going to answer this question using some algebra. So we’ll begin by assigning a letter to the input of our function machines. Let’s call it 𝑥. Let’s then consider what each of these two function machines will do to our input 𝑥.
Function machine A first multiplies our input by six. So multiplying 𝑥 by six will give six 𝑥. We then have to subtract two. So the output of function machine A will be six 𝑥 minus two.
Now, let’s consider function machine B. Remember we have the same input 𝑥. First, we’ll multiply this by two which will give two 𝑥. Then, we need to add eight. So the overall output from function machine B will be two 𝑥 plus eight.
So we’ve now got expressions for the output of each function machine. The question asks us to work out the value of the input such that the output of A is two times or twice the output of B. The output of function machine A is the expression six 𝑥 minus two. And we want this to be twice the output of B; that’s two multiplied by two 𝑥 plus eight.
We now have an equation which we want to solve in order to find the value of 𝑥 which is the input that will make this statement true. We begin by expanding the bracket on the right-hand side. Two multiplied by two 𝑥 gives four 𝑥. And then, two multiplied by positive eight gives positive 16. So we have the equation six 𝑥 minus two equals four 𝑥 plus 16.
Notice that we have terms involving 𝑥 on both sides of this equation. So our next step is going to be to collect all of the 𝑥 terms on the same side. We have a greater number of 𝑥s on the left of the equation. So we’ll collect our terms on this side.
To do so, we need to subtract four 𝑥 from each side of the equation. On the left-hand side, six 𝑥 minus four 𝑥 gives two 𝑥. We bring down the negative two. And on the right-hand side, four 𝑥 minus four 𝑥 leaves no 𝑥s. So we just bring down the positive 16. And we have two 𝑥 minus two equals 16.
Next, we need to add two to each side of this equation. On the left-hand side, two 𝑥 minus two and then plus two just leaves two 𝑥. And on the right-hand side, 16 plus two is 18. The final step in solving this equation is to divide both sides by two. On the left, two 𝑥 divided by two gives 𝑥. And on the right, 18 divided by two gives nine. So we have 𝑥 is equal to nine.
We’ve solved our equation then. But let’s check our answer by substituting this value of nine into each of our function machines. If we substitute nine as our input into function machine A, we first need to multiply it by six. Nine multiplied by six gives 54. We then need to subtract two which gives 52. So the the output from function machine A is 52.
Substituting the same input of nine into function machine B, we first need to multiply by two which gives 18. We then need to add eight which gives 26. So the output from function machine B is 26.
52 is indeed equal to two times 26. So the output of function machine A is twice the output of function machine B when the input is nine. So we’ve checked our answer. And we can say then that our answer to the problem is nine.