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

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

2x + xy + y = 18

x - y = 2

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

x³ + 12xy² = 7x²y

x + y = 20

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

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).

 

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

x² - 4y² = 108

x = 18 - 2y

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

3x - y = 10

x² + 8x - y² + 3y = 17
 

The sum of a number and twice another number is ten less than the product of the numbers. The sum of the numbers is ten. What are all possible numbers that satisfy these criteria?