# Gaussian Elimination Method

### System of Three Equations with Three Unknowns

The **gaussian method** is to use the **elimination method** so that in every equation there is one unknown less than in the previous equation.

1.Place the equation with the coefficient of x**: 1 or −1**, as the first equation. If not possible with x, do with * y* or

*, and change the order of the unknowns:*

**z**2. Carry out the ** elimination method with the 1st and 2nd equation** to

**eliminate**the term of

**x**in the

**2nd equation**. Then, in the second equation, place the result of the operation:

**E' _{2} = E_{2} − 3E_{1} **

3. Do the same with the **1st and 3rd equation** to **eliminate** the term of **x**:

**E' _{3} = E_{3} − 5E_{1}**

4. Perform the elimination method with the 2nd and 3rd equation:

**E'' _{3} = E'_{3} − 2E'_{2} **

5. Another equivalent system is obtained:

6.Solve the system:

z = 1

− y + 4 ·1 = −2 y = 6

x + 6 −1 = 1 x = −4

#### Examples

#### Word Problems

A supermarket customer has paid a total of $156 for 24 litres of milk, 6 kg of ham and 12 litres of olive oil. Calculate the price of each item, knowing that 1 litre of oil costs three times the price of 1 litre of milk and 1 kg of ham costs the same as 4 litres of oil and 4 litres of milk.

milk x

ham y

olive oil z

milk $1

ham $16

olive oil $3

A video store specializes in films in the following three genres: childrens, western and horror. It is known that:

60% of children's movies plus 50% of the westerns account for 30% of all movies.

20% of the childrens, 60% of the westerns and 60% of the horror films account for half of all films in the video store.

There are 100 more western movies than childrens movies.

Find the number of films in each genre.

childrens x

western y

horror z

childrens 500 movies

western 600 movies

horror 900 movies

The sides of a triangle are 26, 28 and 34 cm. Centered at each vertex are three circles, tangent to each other. Calculate the length of each circle's radius.