Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }12~ x&+~~8~ y&+~~9~ z&~=~-300\\7~ x&+~~10~ y&+~~5~ z&~=~1350\\3~ x&-~~~11~ y&-~~~4~ z&~=~280\end{array}$$

Step 2: Multiply the first equation by $ -\frac{ 7 }{ 12 } $ and add the result to the second equation. The result is:

$$\begin{array}{ cccc }12~ x&+~~8~ y&+~~9~ z&~=~-300\\&\dfrac{ 16 }{ 3 }~ y&-~~~\dfrac{ 1 }{ 4 }~ z&~=~1525\\3~ x&-~~~11~ y&-~~~4~ z&~=~280\end{array}$$

Step 3: Multiply the first equation by $ -\frac{ 1 }{ 4 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }12~ x&+~~8~ y&+~~9~ z&~=~-300\\&\dfrac{ 16 }{ 3 }~ y&-~~~\dfrac{ 1 }{ 4 }~ z&~=~1525\\&-~~~13~ y&-~~~\dfrac{ 25 }{ 4 }~ z&~=~355\end{array}$$

Step 4: Get rid of the fractions by multiplying the second equation by 12 and the third equation by 4. The result is:

$$\begin{array}{ cccc }12~ x&+~~8~ y&+~~9~ z&~=~-300\\&64~ y&-~~~3~ z&~=~18300\\&-~~~52~ y&-~~~25~ z&~=~1420\end{array}$$

Step 5: Multiply the second equation by $ \frac{ 13 }{ 16 } $ and add the result to the third equation. The result is:

$$\begin{array}{ cccc }12~ x&+~~8~ y&+~~9~ z&~=~-300\\&64~ y&-~~~3~ z&~=~18300\\&&-~~~\dfrac{ 439 }{ 16 }~ z&~=~\dfrac{ 65155 }{ 4 }\end{array}$$

Step 6: Get rid of the fractions by multiplying the third equation by 16. The result is:

$$\begin{array}{ cccc }12~ x&+~~8~ y&+~~9~ z&~=~-300\\&64~ y&-~~~3~ z&~=~18300\\&&-~~~439~ z&~=~260620\end{array}$$

Step 7: Solve for $ z $.

$$ \begin{aligned} -439 ~ z & = 260620 \\ z & = -\dfrac{ 260620 }{ 439 } \end{aligned} $$

Step 8: Solve for $ y $ using the second equation:

$$ \begin{aligned}64y-3z &= 18300\\64y-3\cdot \left( -\frac{ 260620 }{ 439 } \right) &= 18300\\y &= \dfrac{ 113310 }{ 439 } \end{aligned}$$

Step 9: Solve for $ x $ by substituting $ y = \dfrac{ 113310 }{ 439 } $ and $ z = -\dfrac{ 260620 }{ 439 } $ into the first equation.

System 3x3 calculator