solve the system x+y+z=105 76x+53y+64z=6517 x-z=6622 x=? y=? z=?

Question

solving linear equations with 3 variables by using matrices

Answer 1

Hi Sarah, I have the answer. 

This answer involves big numbers and fractions.  So, stay with me.

 

1     1     1   |   105

76   53   64  |  6517

1     0    -1   |  6622    So, let's begin.

 

 

 

Switch R1 & R3

 

1     0   -1   | 6622

76   53  64  | 6517

1     1    1   |  105

 

-1(R1) + R3  (on R3)

 

1     0   -1   |  6622

76   53  64  |  6517

0     1     2  | -6517

 

-76(R1) + R2  (on R2)

 

1     0    -1   |  6622

0    53  140  | -496755

0     1     2   | -6517

 

-70(R3) + R2  (on R2)

 

1     0    -1   |  6622

0    -17    0  | -40565

0     1     2   | -6517

 

R2/(-17)  on R2

 

1     0    -1   |  6622

0     1     0   |  40565 / 17

0     1     2   | -6517

 

-1(R2) + R3

 

1     0    -1   |  6622

0     1     0   |  40565 / 17

0     0     2   | -151354 / 17

 

R3 / 2

 

1     0    -1   |  6622

0     1     0   |  40565 / 17

0     0     1   | -75677 / 17

 

-1(R3) + R1

 

1     0     0   |  36897 / 17

0     1     0   |  40565 / 17

0     0     1   | -75677 / 17

 

Let's check all the conditions:

x + y + z = 105, 76x + 53y + 64z = 6517, x - z = 6622

 

x + y + z = 105

36897 + 40565 -75677

              17                    

 

1785 = 105

   17 

 

This condition (equation) checks out.

 

76 (36897) + 53 (40565) - 64 (75677)  = 6517

                       17

 

2804172 + 2149945 - 4843328

                    17                           

 

110789 = 6517

    17

 

This condition (equation) checks out.

 

x - z = 6622

 

36897 - (-75677)

          17

 

112574 = 6622

   17      

 

This condition (equation) checks out.  Everything checks out!!!

 

There you have it, Sarah. 

 

x = 36897 / 17,   y = 40565 / 17,   z = -75677 /17.

 

I know the answers are in fractions but it is the answer.

 

I left with fractions and calculate with fractions when I was working through this because it makes easier to add and subtract.

 

Good luck to you.

 

P.S.  A lot of people gave up on this problem because it involves big numbers and fractions.

 

Answer 2

From the look at your equations, we can set up a matrix as follows:

 

1      1       1        105

76   53     64        6517

1      0       1        6622

 

 

Let me know if you understand the setup.

 

Jason D.