Use matrices to find the general solution of the system, if a solution exists

Question

2x - 5y + z =0
x + 4y - 6z =13
3x - 4y -2z =7

The system of equations may have a unique solution, an infinite number of solutions, or no solution. \ (If there is no solution, enter NO SOLUTION. If there are infinitely many solutions, express your answers in terms of z as in Example 3.)
I'm under a time crunch and do not have a graphing calculator. Your help would be soo appreciated!!

Answer 1

 

2x - 5y + z = 0

x + 4y - 6z = 13

3x - 4y -2z =7

 

Add equations 2 and 3

 

4x - 8z = 20 ... 4(x-2z)= 20 ... x-2z = 5

 

Multiply eqn 1 by 4 and eqn 3 by -5 and add:

8x - 20y + 4z = 0

-15x + 20y + 10z = -35

------------------------------

-7x + 14z = -35  ... -7(x - 2z)=-35 ... x-2z = 5

 

This seems to suggest infinite solutions, but

we need to compare equations 1 and 2 to get

another equation with just x and z variables

 

take 4 times equation 1 and 5 times equation 2

and add.

 

8x - 20y + 4z = 0

5x + 20y - 30z = 65

---------------------------

13x - 26z = 65

13(x - 2z) = 65

 x - 2z = 5

 

In similar manner I eliminated the z variable

and in each case got x - 2y = 1

 

Going through and eliminating the x variable

in each case I got y-z=2

 

This system has infinite solutions since we

come up to the same exact equation all three

times when eliminating any one variable.