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# System with a Product

Pro Problems > Math > Algebra > Equations > Systems of Equations > Non-Linear

## System with a Product

Find all ordered pairs (x, y) such that

2x + xy + y = 18

x - y = 2

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## Solution

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## Similar Problems

Mary and Laura Ingalls each receive a cookie. Because they are thoughtful children, they want to share with their little sister Carrie. Because their math skills aren't very advanced, they each eat half of a cookie, but then realize that leaves a full cookie for Carrie.

How much should each girl eat in order to share equally among the three sisters?

### One Equation, Two Variables

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

### Cubic and Linear

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

x3 + 12xy2 = 7x2y

x + y = 20

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)

### Sum and Product System

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?

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

3x - y = 10

x2 + 8x - y2 + 3y = 17

The sum of two numbers is seven times the difference between three times the second number and twice the first number. If the second number is subtracted from the first, the result is the square root of the first. Find all possible values for the first number.

### Product of X and Y

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).

### X and Y System

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

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

x2 - 4y2 = 108

x = 18 - 2y

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