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.

Gaussian Elimination Method

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 z, and change the order of the unknowns:

Gaussian Elimination Method

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 = E2 − 3E1

Gaussian Elimination Method

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

E'3 = E3 − 5E1

Gaussian Elimination Method

Gaussian Elimination Method

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

E''3 = E'3 − 2E'2

Gaussian Elimination Method

5. Another equivalent system is obtained:

Gaussian Elimination Method

6.Solve the system:

z = 1

− y + 4 ·1 = −2        y = 6

x + 6 −1 = 1          x = −4


Examples

System of Equations

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

System of Equations Solution

System of Equations Solution


System of Equations

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

Gaussian Elimination Method

System of Equations Solution

System of Equations Solution


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 flecha x

ham flecha y

olive oil flecha z

System of Equations

Gaussian Elimination Method

Gaussian Elimination Method

System Solution

milk flecha $1

hamflecha $16

olive oil flecha $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.

childrensflecha x

westernflecha y

horror flecha z

System of Equations

System of Equations

System of Equations

System Solution

childrens flecha 500 movies

westernflecha 600 movies

horror flecha 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.

Circles at Triangular Vertices

System of Equations

System of Equations

System of Equations

System Solution

System Solution