Solve the linear system by gaussian-jordan elimination. a) x1 + x2 + 2x3 = 8 b) x – y + 2z - w = -1 – x1 – 2x2 + 3x3 = 1 2x + y - 22 – 2w = -2 3x1 – 7x2 + 4x3 = 10 -x + 2y – 4z +w=1 3x – 3w = -3

Solution for the linear system:

a)  

b)

z and w are free, meaning that can have any value, for this reason, this system has infinite solutions.

Step-by-step explanation:

Gaussian-Jordan elimination consists of taking an augmented matrix, and transform it into its Row echelon form by means of row operation.  For notation, R_i will be the transform column, and r_i the actual one.

Linear System a)

First, you have to convert the system into matrix notation, in this case, column 1 corresponds to variable x_1, column 2 to x_2, column 3 to x_3 and column 4 to the system constants:

Transform to 0 every number in the rows under the first row.

Operations:

Transform the -1 of the second row into 1

Operations:

Transform to 0 the numbers under the 1 in the second row

Operations:

Transform the -52 in the row 3 into 1

Operations:

Now write the matrix like linear equations

Now that you know the valuo of x_3, you can solve the system starting from the bottom up, then find x_2 and finally x_1

Linear System b)

For this system, the process is the same as the above.  

Convert the system into matrix form

Operations:

Operations:

Operations:

Now you can write the system as equations:

For w and z there is no unique answer, so the system result is expressed in terms of those variables. This system has infinite solutions.

Solution:

z and w are free values.

Set up a system of equations

8,400-6,400=xx/6,400anyways the percent increase is 31.25% or 0.3125 you can round if you can