Question

Find if no solution, only one solution and if so, find the variables, or infinitely many solutions?
-4x+6y-5z=-5
-2x+5y-3z=1
3x-2y+3z=3

Answer 1

the 1st task is to reduce the 3 equations to 2 equations with 2 unknowns


Equation number
-4x + 6y - 5z = -5 (1)
-2x + 5y - 3z = 1 (2)
3x - 2y + 3z = 3 (3)

I'd choose to eliminate x so its equation 1 + -2 * equation 2

-4x + 6y - 5z = -5
4x - 10y +6z = -2
-------------------
-4y + z = -7 (4)


next 3* equation (2) + 2* equation (3)

-6x + 15y - 9z = 3
6x - 4y + 6z = 6
--------------------

11y -3z = 9 (5)

now you have 2 equations

-4y + z = -7 (4)
11y - 3z = 9 (5)

eliminate z 3*equation (4) + equation (5)

-12y + 3z = -21
11 y - 3z = 9

then -y = -12 or y = 12

substitute into equation (4) to find z

-4(12) + z = -7

- 48 + z = -7

z = 41

now with y =12 and z = 41 substitute into equation 1 to find x

-4x + 6(12) - 5(41) = -5

-4x +72 - 205 = -5

x = -32

so the solution is x = -32, y = 12 and z = 41