# Video: Writing the Rule of an Input-Output Table as an Equation

In this function machine, we put 𝑥 in and get 𝑦 out. It uses the rule add 20 to 𝑥 to calculate 𝑦. If we put 3 in, what number do we get out? If we get out 32, what number did we put in? Which of these equations describes the rule? [A] 𝑦 = 𝑥 − 23 [B] 𝑦 = 𝑥 + 23 [C] 𝑦 = 𝑥 × 20 [D] 𝑦 = 𝑥 − 20 [E] 𝑦 = 𝑥 + 20.

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### Video Transcript

In this function machine, we put 𝑥 in and get 𝑦 out. It uses the rule add 20 to 𝑥 to calculate 𝑦. Using this machine, we’ll try to answer three questions. If we put three in, what number do we get out? If we get out 32, what number did we put in? And which of these equations describes the rule?

Starting with our first question, if we put three in, what number do we get out? First, we examine our table to see if it tells us what the output for three would be. Three is not listed in our table, which means we’ll have to calculate the output ourselves. If we put in three for our 𝑥-value, we know that we add 20 to our 𝑥-value to find the 𝑦. What is three plus 20? 23 If we put three in, we will get 23 out.

Our next question is slightly different. It says if we get out 32, what number did we put in? First, we examine our table to see if any of the output values listed are 32. Are any of the 𝑦-values listed 32? Since there was not an output value of 32 in the table, we added 32 to the 𝑦 row. And now, we need to find out what input — what 𝑥-value — would give us 32.

We know moving from 𝑥 to 𝑦 is plus 20. How would we move in the opposite direction? What operation is the opposite of plus 20? Moving this direction, doing the opposite of adding 20, is subtracting 20 — taking 20 away. If we have 32 as our output value, we can subtract 20 to find the input value. 32 minus 20 equals 12. If we get out 32, we put 12 in.

Our last question wants us to take the rule for this function machine and turn it into an equation. The rule for this function machine is to add 20. Any of our answer choices that have subtraction, division, or multiplication cannot be the correct equation.

By ruling out all the equations that do not have addition, we’ve narrowed it down to just two: 𝑦 equals 𝑥 plus 23 or 𝑦 equals 𝑥 plus 20. But going back to our original rule, we know that we’re adding 20 every time. And only one of these equations represents 𝑥 plus 20.

The equation for this function machine is 𝑦 equals 𝑥 plus 20.