Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }3~ x&&&~=~12\\~~ x&+~~~~ y&&~=~5\\&-~~~~~ y&+~~~~ z&~=~99\end{array}$$

Step 2: Find the determinant of the coefficient matrix.

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

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.

$$ D_x= \left| \begin{array}{ ccc }\color{black}{ 12 }&\color{blue}{ ~~0 }&\color{green}{ ~~0 }\\\color{black}{ 5 }&\color{blue}{ ~~1 }&\color{green}{ ~~0 }\\\color{black}{ 99 }&\color{blue}{ -1 }&\color{green}{ ~~1 }\end{array} \right| = 12 $$

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

$$ D_y= \left| \begin{array}{ ccc }\color{red}{ 3 }&\color{black}{ ~~12 }&\color{green}{ ~~0 }\\\color{red}{ 1 }&\color{black}{ ~~5 }&\color{green}{ ~~0 }\\\color{red}{ 0 }&\color{black}{ ~~99 }&\color{green}{ ~~1 }\end{array} \right| = 3 $$

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

$$ D_z= \left| \begin{array}{ ccc }\color{red}{ 3 }&\color{blue}{ ~~0 }&\color{black}{ ~~12 }\\\color{red}{ 1 }&\color{blue}{ ~~1 }&\color{black}{ ~~5 }\\\color{red}{ 0 }&\color{blue}{ -1 }&\color{black}{ ~~99 }\end{array} \right| = 300 $$

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:

$$ \begin{aligned} x &= \frac{D_x}{D} = \frac{ 12 }{ 3 } = 4 \\y &= \frac{D_y}{D} = \frac{ 3 }{ 3 } = 1 \\z &= \frac{D_z}{D} = \frac{ 300 }{ 3 } = 100 \end{aligned} $$

System 3x3 calculator