Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }-3~ x&+~~~~ y&&~=~-10\\~~ x&-~~~9~ y&+~~3~ z&~=~0\\&3~ y&-~~~5~ z&~=~5\end{array}$$

Step 2: Find the determinant of the coefficient matrix.

$$ D= \left| \begin{array}{ ccc }\color{red}{ -3 }&\color{blue}{ ~~1 }&\color{green}{ ~~0 }\\\color{red}{ 1 }&\color{blue}{ -9 }&\color{green}{ ~~3 }\\\color{red}{ 0 }&\color{blue}{ ~~3 }&\color{green}{ -5 }\end{array} \right| = -103 $$

Step 3: Find the determinant of the x-matrix (D_x). X-matrix is formed by replacing the x-column values with the answer-column values.

$$ D_x= \left| \begin{array}{ ccc }\color{black}{ -10 }&\color{blue}{ ~~1 }&\color{green}{ ~~0 }\\\color{black}{ 0 }&\color{blue}{ -9 }&\color{green}{ ~~3 }\\\color{black}{ 5 }&\color{blue}{ ~~3 }&\color{green}{ -5 }\end{array} \right| = -345 $$

Step4: Find the determinant of the y-matrix. Y-matrix is formed by replacing the y-column with the constant column.

$$ D_y= \left| \begin{array}{ ccc }\color{red}{ -3 }&\color{black}{ -10 }&\color{green}{ ~~0 }\\\color{red}{ 1 }&\color{black}{ ~~0 }&\color{green}{ ~~3 }\\\color{red}{ 0 }&\color{black}{ ~~5 }&\color{green}{ -5 }\end{array} \right| = -5 $$

Step 5: Find the determinant of the z-matrix.

$$ D_z= \left| \begin{array}{ ccc }\color{red}{ -3 }&\color{blue}{ ~~1 }&\color{black}{ -10 }\\\color{red}{ 1 }&\color{blue}{ -9 }&\color{black}{ ~~0 }\\\color{red}{ 0 }&\color{blue}{ ~~3 }&\color{black}{ ~~5 }\end{array} \right| = 100 $$

Cramers Rule says that the solutions are

$$ x = \frac{D_x}{D} ~,~y = \frac{D_y}{D}~,~z = \frac{D_z}{D} $$

So, in this example we have:

$$ \begin{aligned} x &= \frac{D_x}{D} = \frac{ -345 }{ -103 } = \frac{ 345 }{ 103 } \\y &= \frac{D_y}{D} = \frac{ -5 }{ -103 } = \frac{ 5 }{ 103 } \\z &= \frac{D_z}{D} = \frac{ 100 }{ -103 } = -\frac{ 100 }{ 103 } \end{aligned} $$

System 3x3 calculator