   SEARCH HOME Math Central Quandaries & Queries  Question from Karlena, a student: This is a question from the math class that I want to get into in college so please help me out. I am supposed to write the augmented matrix of the system and use the matrix method to solve the system. I must show my work algebraically x+y+2z=30 2x+3y+2z=53 x+2y+3z=47 Hi Karlena,

The augmented matrix of a linear system of equations contains the coefficients of the variables augmented by a column for the constants. For your system the augmented matrix is

 1 1 2 30 2 3 2 53 1 2 3 47

(I have written the matrix as a table as it is much easier for me to manipulate a table than the standard matrix notation.)

The task now is to perform elementary row operations to transform the matrix to have zeros below the diagonal and if preferably with 1 as the diagonal value in each row. (The diagonal of interest starts at the upper left corner so currently the diagonal entries are 1, 3 nd 3.) There are many choices for row operation but here is what I did.

Multiply the first row by -2 and add it to the second row. I get

 1 1 2 30 0 1 -2 -7 1 2 3 47

Multiply the first row by -1 and add it to the third row. I get

 1 1 2 30 0 1 -2 -7 0 1 1 17

Multiply the second row by -1 and add it to the third row. I get

 1 1 2 30 0 1 -2 -7 0 0 3 24

Divide the third row by 3. I get

 1 1 2 30 0 1 -2 -7 0 0 1 8

At this point rewrite the system as equations rather than in matrix form.

 x + y + 2z = 30 y - 2z = -7 z = 8

The last equation tells me z = 8. Substitute z = 8 into the second equation and solve for y. Substitute this value for y and z = 8 into the first equation and solve for x.     