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# Sum and Product System

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

## 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?

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Problem by allie

## Solution

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## Similar Problems

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

x2 - 4y2 = 108

x = 18 - 2y

### 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

### System with a Product

Find all ordered pairs (x, y) such that

2x + xy + y = 18

x - y = 2

### 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).

### Sum of X and Y

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

### To Sum It Up

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?

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?

Solve for m and n.

(m + n)2 - 10(m + n) + 24  = 0

(m - n)2 + 6(m - n) + 8 = 0   Like us on Facebook to get updates about new resources