The sum of two numbers is 47 and the difference is 25 . What are the numbers?

Answer 1

The question is asking for two numbers, so we need to designate two variables. In this example, I will use x and y. The first part of the problem states that the two numbers added together equals 47, so we write: x + y = 47. The second part says their difference is 25, so we subtract y from x to get 25, or x - y = 25.

How do we figure out our numbers? We use a system of equations, which are the two we created above:

x + y = 47

x - y = 25

The next step will be to eliminate one of the variables. We do this by finding how we can combine the equations to remove a variable. Since we have a positive and negative y in our equations, we simply add the two equations together.

x + y = 47

+ x - y = 25

The result is: x + x + y - y = 47 + 25, which simplifies to: 2x = 72. Dividing both sides by two gives us: x = 36.

The next step is to substitute our solution, 36, for x into one of the equations. We will use the first one: 36 + y = 47. Subtracting 36 from both sides gives us y = 11, so our final answer to the question is: 36 and 11.

Answer 2

In this case, we know we have two numbers which are unknown to us. We can call those by two different names, let's call them "x" and "y". We know that together, they add together, or "sum" to make 47. Using this information to set up an equation, we would say that x + y = 47 (equation 1)

We also know that the difference is 25, meaning that one number subtracted from the other gives us 25. It is not important which name represents the greater number. Let's say that x is larger, so our second equation would be x - y = 25 (equation 2).

Now we can solve for each of our unknown numbers, or "variables". We do so by isolating one at a time, and then plugging it in to the other equation. Our goal is to create an equation in which only one of our variables (x or y) is present, so that we can solve for it's number.

In the case of equation 1, we can subtract "y" from either side to get "x" alone on one side. This will give us the equation x = 47-y. Subtracting "y" from the left side of the equation will help us to isolate "x", and we must subtract y on both sides of the equation to ensure it is even.

Now, we can plug in this value for x into equation two. Doing so gives us the following equation: (47-y) - y = 25. Now that we have an equation with only one variable present, "y", we can solve for y. In this case, our two "-y"'s can combine to give us 47 - 2y = 25. We now need to isolate "y" by itself. Subtracting 47 from either side will give us -2y = 25-47 = -22. Now that we know that -2y = -22, we can divide both sides by "-2" to see that y = 11.

Now that we know the identity of our variable "y", we can plug it in to either equation 1 or equation 2 to solve for the identity of our variable "x". If we plug 11 in for "y" in equation 1, we can see that x + 11 = 47. Subtracting 11 from each side (to isolate x) lets us see that x = 36.

Now that we have solved that x = 36 and y = 11, we should double check using equation 2 that we got the correct answer. If this is the case, x - y should equal 25. And it does! 36-11 = 25, which helps us confirm that our equations were solved correctly.