Full solution

3

x -

3

y + z =

-3

(\newline\)

-3

x +

3

y - z =

3

(\newline\)

-2

x +

3

y +

3

z =

15

Eliminate z: Add the first and second equations to eliminate $z$ . $\newline$ $(3x - 3y + z) + (-3x + 3y - z) = -3 + 3$ $\newline$ $0 = 0$
Eliminate x: Add the first and third equations to eliminate $x$ . $\newline$ $(3x - 3y + z) + (-2x + 3y + 3z) = -3 + 15$ $\newline$ $x + 4z = 12$
Eliminate x: Add the second and third equations to eliminate $x$ . $\newline$ $(-3x + 3y - z) + (-2x + 3y + 3z) = 3 + 15$ $\newline$ $-5x + 6y + 2z = 18$
Solve for x: Solve for $x$ from $x + 4z = 12$ . $\newline$ $x = 12 - 4z$
Solve for y: Substitute $x = 12 - 4z$ into $-5x + 6y + 2z = 18$ . $\newline$ $-5(12 - 4z) + 6y + 2z = 18$ $\newline$ $-60 + 20z + 6y + 2z = 18$ $\newline$ $22z + 6y = 78$ $\newline$ $11z + 3y = 39$
Substitute x and y: Solve for $y$ from $11z + 3y = 39$ . $\newline$ $3y = 39 - 11z$ $\newline$ $y = \frac{39 - 11z}{3}$
Check in first equation: Substitute $x = 12 - 4z$ and $y = \frac{39 - 11z}{3}$ into the first equation $3x - 3y + z = -3$ . $\newline$ $3(12 - 4z) - 3\left(\frac{39 - 11z}{3}\right) + z = -3$ $\newline$ $36 - 12z - 39 + 11z + z = -3$ $\newline$ $36 - 39 = -3$ $\newline$ $-3 = -3$
Check in second equation: Check the solution in the second equation $-3x + 3y - z = 3$ . $\newline$ $-3(12 - 4z) + 3\left(\frac{39 - 11z}{3}\right) - z = 3$ $\newline$ $-36 + 12z + 39 - 11z - z = 3$ $\newline$ $3 = 3$
Check in third equation: Check the solution in the third equation $-2x + 3y + 3z = 15$ . $\newline$ $-2(12 - 4z) + 3\left(\frac{39 - 11z}{3}\right) + 3z = 15$ $\newline$ $-24 + 8z + 39 - 11z + 3z = 15$ $\newline$ $15 = 15$