𝑥 plus two 𝑦 equals 20. 𝑥 and 𝑦 are whole numbers less than 10. What could 𝑥 and 𝑦 be?
This problem starts off with an expression, where we don’t know two of the numbers. We start off with a number. Then, we add two lots of another number. And we get the answer 20. The missing numbers that we don’t know are represented by letters 𝑥 and 𝑦. And the only thing that we know about these numbers apart from the number sentence itself is that they’re both whole numbers less than 10. The question asks us to find what 𝑥 and 𝑦 could be.
Notice the word “could” in the answer. What could they be? This is slightly different than asking what are 𝑥 and 𝑦. When we get a question that asks what could something be, it generally means there might be more than one answer. We only need to give one answer. But they could be more than one possibility. Let’s start by looking at this expression. We could draw a bar model to help us understand it.
We start off with 𝑥 and then we add two 𝑦. Remember when we have a number next to a letter like this, it means two times 𝑦, two lots of 𝑦. 𝑥 plus two lots of 𝑦 equals 20. And so, the whole bar has a value of 20. There’s actually one more thing that we know for sure this time about the number that 𝑥 represents. Is it odd or even? We know 20 is an even number. And two lots of 𝑦 must also be an even number. It doesn’t matter whether 𝑦 is even or odd. Two lots of it will make it an even number. So if we take away an even number from an even number, we get an even number.
So although we don’t know whether 𝑦 is odd or even, we do know that 𝑥 is an even number. So we won’t bother trying the number one. 𝑥 can’t be one because one is odd. Let’s try two. What if 𝑥 equals two? If 𝑥 equals two, then our expression becomes two plus two lots of something equals 20. What do we add to two to make 20? Well, two and 18 go together to make 20. So two lots of 𝑦 must equal 18. 18 split in half equals nine. And so, 𝑥 could be equal to two and 𝑦 could be equal to nine.
We did say there was more than one possible answer. So let’s go through and see which other answers we could have put. We said that if 𝑥 equals two, 𝑦 must be equal to nine. Let’s think about the next even number. That’s a whole number less than 10. Four, what if 𝑥 was equal to four? If 𝑥 equals four, then four plus two lots of something equal 20. What goes together with four to make 20? Four plus 16 make 20. So two lots of 𝑦 must equal 16. And if we divide 16 by two to find one lot of 𝑦, we get the answer eight. So another possible combination that we could give us the answer is 𝑥 equals four, 𝑦 equals eight.
The next even whole number is six. What if we found an answer where 𝑥 equals six? Well, where 𝑥 equals six, we could write six plus two lots of 𝑦 equal 20. Again, we need to think what number goes with six to make 20. Six plus 14 make 20. So two lots of 𝑦 must equal 14. We need to divide that number then by two to find one lot of 𝑦. If 𝑥 is equal to six, then 𝑦 is equal to seven. And we could write those two numbers as a possible answer. It seems to be a pattern here in the 𝑥 and 𝑦 numbers.
Let’s make a prediction about the last two possible answers that we could give and see if we’re right. The 𝑥 numbers, the possible answers for 𝑥, seem to go up in twos: two, four, six. And we know the last even whole number less than 10 is eight. And if we look at the 𝑦 answers, nine, eight, seven, we could predict that the next number could be six.
Let’s see if we’re right. If 𝑥 equals eight, then our expression becomes eight plus two 𝑦 equals 20. What do we add to eight to make 20? Eight plus 12 equals 20. So now we know that two lots of 𝑦 equal 12, we can divide by two to find one lot of 𝑦. 12 divided by two equals six. So another possible pair of answers is 𝑥 equals eight, 𝑦 equals six.
There are no more possible answers. Because as we move upwards, 𝑥 will become 10 and more. And we’re told the 𝑥 is a whole number less than 10. 𝑥 plus two 𝑦 equals 20. 𝑥 and 𝑦 are whole numbers less than 10. And the possible answers that 𝑥 and 𝑦 could be are 𝑥 equals two, 𝑦 equals nine. That’s the answer that we’ve given. But there are also three other possible answers that you could give. One is 𝑥 equals four, 𝑦 equals eight. Another is 𝑥 equals six, 𝑦 equals seven. And the final possible answer we could have written is 𝑥 equals eight, 𝑦 equals six.