Question 695573: The sum of 2 numbers is 17. The second number is 3 more than the first. What are the two numbers? Write an equation for the question.
Answer by RedemptiveMath(80)   (Show Source):
You can put this solution on YOUR website! For these problems, we need to convert regular English language into mathematical language. The difference between the two is that the latter is written in almost all mathematical symbols and operations. To begin working on this problem, we need to know what key words in English relate to their respective operations in mathematics. The first sentence says, "The sum of 2 numbers is 17." The first math word we come across is "sum". We know that the sum is the addition of two or more numbers. If we read futher, we know that we are adding two numbers together. Since we do not know these numbers yet, we can begin writing our first "mathematical sentence":
x + y...
It's good if we use two different variables because we are dealing with two separate numbers. (If the two numbers were equal, which would make us only deal with one variable, then the information wouldn't have stated that there were two numbers. If the two numbers were equal, they would be the same number and thus not two numbers.) "X + y" is mathematical language for "the sum of two numbers." Now let us finish the rest of our mathematical sentence by reading the rest of the first English sentence. The last two terms of the first English sentence let's us know that the sum is 17. "Is" or "are" (depending on if we're dealing with single or multiple things) are the keywords in English that let us know we are talking about "equal" in mathematics. So, let us finish our first mathematical sentence:
x + y = 17.
We have gotten the statement above by converting English language into mathematical language. This is our first equation. We still have more to analyze, namely the second sentence of information in the problem. Let us continue.
"The second number is 3 more than the first." Again, we must convert this English language into mathematical language. When we do so, we'll get our second equation. Then we can begin to solve. "The second number" can realistically be either of the two variables we've chosen (as long as we know which is first and which is second as we move along), but for our sake let us choose the variable y to represent the second number. This sentence says that the second number, or y, is 3 more than the first number, which we have made the other variable x. "More" is a keyword in English that lets us know we are dealing with addition in mathematical language. So, let us write our second mathematical sentence:
y = 3 + x (remember that "is" means equals)
y = x + 3 (we can rewrite the order of addition on the right side of the equation if we'd like).
Now we have our two equations. We can use the methods to solve systems of linear equations (for this is a system) to solve for the variables. Let's take the method of substitution which is simpler in this case.
x + y = 17
y = x + 3
x + (x + 3) = 17 (substitute y = x + 3 into the first equation)
2x + 3 = 17 (combine like terms)
2x = 14 (subtracting 3 from both sides)
x = 7 (divide by 2 on both sides).
We have found our first number! If you chose x to be the greater of the two numbers, then you'd have solved for y first. Now all we have to do is plug this number into either equation:
x + y = 17
7 + y = 17 (substitute x = 7 into the equation)
y = 10 (subtract 7 from both sides)
OR
y = x + 3
y = 7 + 3 (substitute x = 7 into the equation)
y = 10 (combine like terms).
As we can see, we get y = 10 for both equations when x = 7. Let us see if 7 and 10 match the requirements for the two numbers we need.
Are they two numbers? Yes.
Is the sum between the two numbers 17? 10 + 7 = 17. Yes.
Is one of the numbers 3 more than the other? 10 = 7 + 3. Yes.
We have found our two numbers. They should be 7 and 10. Your equations would be x + y = 17 and y = x + 3 (or x = y + 3 if you chose x to be greater).
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