the sum of two numbers is 30. the difference of their squares is 180. find the two numbers by elimination method

Let x = one number and y = second number

Hint:
Equation #1: x+y=30
Equation #2: x^2 - y^2 = 180

x = 30-y
(30-y)^2-y^2=180
y^2-60y+900-y^2=180
-60y=-720
y=12
x = 30-12
x=18

To solve this problem using the elimination method, we need to set up a system of equations using the given information.

Let's assume the two numbers are "x" and "y".

Given:

1) The sum of two numbers is 30:
x + y = 30 ----(Equation 1)

2) The difference of their squares is 180:
x² - y² = 180 ----(Equation 2)

To solve this system of equations by elimination, we can eliminate one variable by adding or subtracting the equations. In this case, we can eliminate y by multiplying Equation 1 by -1 and adding it to Equation 2.

Equation 1 multiplied by -1: -x - y = -30

(-x - y) + (x² - y²) = -30 + 180
-x - y + x² - y² = 150

Rearranging Equation 2 using the difference of squares:
(x - y)(x + y) = 150

Substituting Equation 1 into Equation 2:
(x - y)(30) = 150

Divide both sides by 30:
(x - y) = 5 ----(Equation 3)

Now, we have a new equation (Equation 3) that represents the difference of the two numbers. We can substitute this equation back into Equation 1 to find the values of x and y.

Substituting Equation 3 into Equation 1:
x + (x - 5) = 30

Simplifying:
2x - 5 = 30

Adding 5 to both sides:
2x = 35

Dividing both sides by 2:
x = 17.5

Now, we can substitute the value of x into either Equation 1 or Equation 3 to solve for y.

Using Equation 1:
17.5 + y = 30

Subtracting 17.5 from both sides:
y = 12.5

Therefore, the two numbers are x = 17.5 and y = 12.5.