Step 1: Convert all decimals to fractions

$$\begin{array}{ cccc }-1~ x&+~~\dfrac{ 2 }{ 3 }~ y&-~~~2~ z&~=~2\\5~ x&+~~7~ y&-~~~5~ z&~=~6\\\dfrac{ 1 }{ 4 }~ x&+~~~~ y&-~~~\dfrac{ 1 }{ 2 }~ z&~=~2\end{array}$$

Step 2: Get rid of the fractions by multiplying the first equation by 3 and the third equation by 4. The result is:

$$\begin{array}{ cccc }-3~ x&+~~2~ y&-~~~6~ z&~=~6\\5~ x&+~~7~ y&-~~~5~ z&~=~6\\~~ x&+~~4~ y&-~~~2~ z&~=~8\end{array}$$

Step 3: Find the determinant of the coefficient matrix.

$$ D= \left| \begin{array}{ ccc }\color{red}{ -3 }&\color{blue}{ ~~2 }&\color{green}{ -6 }\\\color{red}{ 5 }&\color{blue}{ ~~7 }&\color{green}{ -5 }\\\color{red}{ 1 }&\color{blue}{ ~~4 }&\color{green}{ -2 }\end{array} \right| = -86 $$

Step 4: Find the determinant of the x-matrix (D_x). X-matrix is formed by replacing the x-column values with the answer-column values.

$$ D_x= \left| \begin{array}{ ccc }\color{black}{ 6 }&\color{blue}{ ~~2 }&\color{green}{ -6 }\\\color{black}{ 6 }&\color{blue}{ ~~7 }&\color{green}{ -5 }\\\color{black}{ 8 }&\color{blue}{ ~~4 }&\color{green}{ -2 }\end{array} \right| = 172 $$

Step5: Find the determinant of the y-matrix. Y-matrix is formed by replacing the y-column with the constant column.

$$ D_y= \left| \begin{array}{ ccc }\color{red}{ -3 }&\color{black}{ ~~6 }&\color{green}{ -6 }\\\color{red}{ 5 }&\color{black}{ ~~6 }&\color{green}{ -5 }\\\color{red}{ 1 }&\color{black}{ ~~8 }&\color{green}{ -2 }\end{array} \right| = -258 $$

Step 6: Find the determinant of the z-matrix.

$$ D_z= \left| \begin{array}{ ccc }\color{red}{ -3 }&\color{blue}{ ~~2 }&\color{black}{ ~~6 }\\\color{red}{ 5 }&\color{blue}{ ~~7 }&\color{black}{ ~~6 }\\\color{red}{ 1 }&\color{blue}{ ~~4 }&\color{black}{ ~~8 }\end{array} \right| = -86 $$

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:

$$ \begin{aligned} x &= \frac{D_x}{D} = \frac{ 172 }{ -86 } = -2 \\y &= \frac{D_y}{D} = \frac{ -258 }{ -86 } = 3 \\z &= \frac{D_z}{D} = \frac{ -86 }{ -86 } = 1 \end{aligned} $$

System 3x3 calculator