The sum of two numbers is 20. The difference of their squares is 120. What are the numbers?

a + b = 20

a^2-b^2 = 120
(a+b)(a-b) = 120
20(a-b) = 120
a-b = 6
so

a+b = 20
a-b = 6
I bet you can solve that

Well, these numbers must be quite interesting characters. Let's give them some names. Let's call one number Mr. X and the other number Mrs. Y.

So, according to the information, Mr. X + Mrs. Y = 20 and Mrs. Y² - Mr. X² = 120.

Now, let's put our detective hats on and solve this mystery!

If we subtract the second equation from the first equation, we get:

2(Mr. X) = -100.

This means that Mr. X = -50.

But hey, it takes two to tango! So if Mr. X is -50, we substitute it back into the first equation to find Mrs. Y:

-50 + Mrs.Y = 20,

therefore Mrs. Y = 70.

So the numbers are Mr. X = -50 and Mrs. Y = 70.

Now, let's hope that they get along better than their equation counterparts did!

To find the two numbers, let's assign variables to represent them. Let's say the first number is 'x' and the second number is 'y'.

According to the problem, the sum of the two numbers is 20, so we can write the equation:

x + y = 20 --- (Equation 1)

The problem also states that the difference of their squares is 120, which means we need to subtract the square of one number from the square of the other. Mathematically, we can express this as:

x^2 - y^2 = 120 --- (Equation 2)

Now, we have a system of two equations (Equation 1 and Equation 2) with two variables (x and y) that we need to solve.

To solve this system, we can apply the method of substitution or elimination. Let's use substitution.

From Equation 1, we can isolate one variable and substitute it into Equation 2.

Solving Equation 1 for x, we get:

x = 20 - y

Now, we can substitute this value of x into Equation 2:

(20 - y)^2 - y^2 = 120

Expanding the equation:

(400 - 40y + y^2) - y^2 = 120

Simplifying and rearranging the terms:

400 - 40y = 120

Subtracting 400 from both sides:

-40y = -280

Dividing both sides by -40:

y = 7

Now that we have the value of y, we can substitute it back into Equation 1 to find x:

x + 7 = 20

Subtracting 7 from both sides:

x = 13

Therefore, the two numbers are 13 and 7.

y=7 and x=9

X+Y=20 OR X=20-Y

X^2-Y^2=120
(20-Y)^2-Y^2=120
400-40Y+Y^2=120
-40Y=120-400
-40Y=-280
Y=-280/-40

Y=7 ans.
X+7=20
X=20-7
X=13 ans.
Proof:
13^2-7^2=120
169-49=120
120=120