The sum of two numbers is 19, and their difference is 55. What are the two numbers?
Solving systems using elimination
x+y = 19
x-y = 55
add them
2x = 74
x = 37
in 1st ---> 37+y=19
y = -18
the numbers are -18 and 37
Well, this is quite a tricky situation! It sounds like these two numbers have a serious case of being opposites. It's like they're trying to play a game of hide and seek, but they just can't seem to find each other!
Let's call the first number "x" and the second number "y". According to the information given, we know that the sum of the two numbers is 19 (x + y = 19) and their difference is 55 (x - y = 55).
Now, I don't know about you, but these numbers seem to be in quite a pickle. How can they have a difference of 55 and a sum of 19 at the same time? They must be playing some kind of mathematical magic trick on us!
Since these numbers are having a difficult time cooperating, I'm afraid there might not be a real solution to this puzzle. It seems like this question is trying to pull off a disappearing act worthy of the circus!
To solve this problem, let's assign variables to the two unknown numbers. Let's call one of the numbers x and the other number y.
From the problem, we know two things:
1. The sum of the two numbers is 19. In equation form, this is x + y = 19.
2. The difference between the two numbers is 55. In equation form, this is x - y = 55.
To solve this system of equations using elimination, we'll eliminate one of the variables by adding or subtracting the two equations. Let's eliminate y.
First, let's multiply the second equation by -1:
-1(x - y) = -1(55)
- x + y = -55
Now, we can add this modified equation to the first equation:
x + y + ( -x + y) = 19 + ( -55)
2y = -36
Next, divide both sides of the equation by 2 to solve for y:
2y/2 = -36/2
y = -18
Now, substitute the value of y (-18) into one of the original equations. Let's use the first equation:
x + (-18) = 19
x - 18 = 19
Add 18 to both sides of the equation to solve for x:
x - 18 + 18 = 19 + 18
x = 37
So, the two numbers are x = 37 and y = -18.
To solve the system of equations, which are the sum and difference of the two unknown numbers, we can use the method of elimination. Here's how we can do it step by step:
1. Assign variables: Let's assign variables to the two unknown numbers. Let's call the first number x and the second number y.
2. Write equations: Based on the given information, we can write two equations:
- The sum equation: x + y = 19
- The difference equation: x - y = 55
3. Multiply equations (optional step): Since the coefficients of y in both equations are already the same (+1 and -1), we don't need to multiply any equation to make their coefficients match.
4. Add equations: Add the two equations together:
- (x + y) + (x - y) = 19 + 55
- 2x = 74
5. Solve for x: Divide both sides of the equation by 2 to solve for x:
- 2x/2 = 74/2
- x = 37
6. Substitute x value: Substitute the value of x back into one of the original equations (either equation will work). Let's use the sum equation:
- 37 + y = 19
7. Solve for y: Solve the equation for y:
- y = 19 - 37
- y = -18
8. Check the solution: Verify that the values of x and y satisfy both original equations:
- x + y = 37 + (-18) = 19 (satisfied)
- x - y = 37 - (-18) = 55 (satisfied)
Therefore, the two numbers are 37 and -18.