Use Gaussian elimination to solve the system 2

Mackenzie Rios

Mackenzie Rios

Answered question

2022-05-28

Use Gaussian elimination to solve the system
2 x 3 y = 5
3 x + y = 9
Find the value of x.

Answer & Explanation

Syllingbs

Syllingbs

Beginner2022-05-29Added 11 answers

Since we want to find the value of x, we need to eliminate y from both the equations i.e. from the equations
2 x 3 y = 5 Equation  1 3 x + y = 9 Equation  2
Though you can do it directly, without resorting to matrices, to enable you to understand how to work with matrices, I am going to do this in the matrix form.
The two equations can be written in a matrix form as shown below.
[ 2 3 3 1 ] [ x y ] = [ 5 9 ]
Typically, Gauss elimination ( L U factorization) involves making the entries below the main diagonal zero using row-operations i.e.
[ 2 3 3 1 ] [ × × 0 × ]
However, you could also do Gauss elimination by making the entries above the main diagonal zero using row-operations i.e.
[ 2 3 3 1 ] [ × 0 × × ]
Since you are interested in x, it makes sense to proceed along the second line and convert
[ 2 3 3 1 ]
to a lower triangular matrix.
This can be done as follows. If we denote the first equation as R 1 and the second equation as R 2 , then the operation R 1 = R 1 + 3 R 2 converts
[ 2 3 3 1 ] [ x y ] = [ 5 9 ]
to
[ 2 + 3 × 3 3 + 3 × 1 3 1 ] [ x y ] = [ 5 + 3 × 9 9 ]  i.e.  [ 11 0 3 1 ] [ x y ] = [ 22 9 ]
The first row/equation can now be used to read of the value for x.
11 x = 22 x = 2
If you are interested in y as well, plug in the value for x and obtain y from the second equation, i.e.
3 × 2 + y = 9 y = 3
Hugo Brady

Hugo Brady

Beginner2022-05-30Added 3 answers

I had the same problem, thanks!

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get an expert answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?