I have picked three positive integers for the lottery, as follows: The sum of my numbers is 54. The sum of my numbers, plus the sum of two of my numbers, is 84. The sum of the squares of my numbers is 1034. What are the three integers?

Find all ordered pairs (x, y) such that:

3x - y = 10

x² + 8x - y² + 3y = 17
 

Find the sum of x and y, if the following are true:

(x + 2)(x - 1) = (y - 12)(y + 3)

(x + 1)(x + 3) = (y - 5)(y - 7)

Find all ordered pairs (x,y) which solve the following non-linear system of equations.

x(x - 2y) - 4 = 2y(x - 2y)

x + 2y = 10

Solve for m and n.

(m + n)² - 10(m + n) + 24  = 0

(m - n)² + 6(m - n) + 8 = 0

Find all ordered pairs (x, y) such that the following two equations are true:

x² - 4y² = 108

x = 18 - 2y

For the ordered pair (x,y) the product of x and y is 108. If x + 2y = 30, find all possible ordered pairs (x,y).

 

Find all ordered pairs (x,y) which solve the following system of equations:

x³ + 12xy² = 7x²y

x + y = 20

Usually we say that if we have two variables, we need two equations to solve, and if we have three variables, we need three equations to solve. This is not 100% true, however, and the problem below is a good example of a single equation in two variables which produce a single ordered pair solution.

Solve for the ordered pair (x,y) such that x² + y² - 2x + 4y = -5

Find the sum of x and y if x and y are positive numbers such that x² + 3xy + y² = 424 and xy = 100