,

คูณตลอดแต่ละสมการด้วย $\log 2$, $\log 3$ และ $\log 4$

$$\begin{eqnarray*} \log x+\frac{1}{2}\log y+\frac{1}{2}\log z & = & 2\log2\\ \frac{1}{2}\log x+\log y+\frac{1}{2}\log z & = & 2\log3\\ \frac{1}{2}\log x+\frac{1}{2}\log y+\log z & = & 2\log4 \end{eqnarray*}$$

คูณด้วยสอง ตลอดทุกสมการ

$$\begin{eqnarray*} 2\log x+\log y+\log z & = & 4\log2\\ \log x+2\log y+\log z & = & 4\log3\\ \log x+\log y+2\log z & = & 4\log4 \end{eqnarray*}$$

,

$$\left[\begin{array}{ccc} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end{array}\right]\left[\begin{array}{c} \log x\\ \log y\\ \log z \end{array}\right]=\left[\begin{array}{c} 4\log2\\ 4\log3\\ 8\log2 \end{array}\right]$$

,

$$\left[\begin{array}{ccc} 2 & 1 & 1\\ 1 & 2 & 1\\ 1 & 1 & 2 \end{array}\left|\begin{array}{c} 4\log2\\ 4\log3\\ 8\log2 \end{array}\right.\right]$$

$$\overrightarrow{-\frac{1}{2}R_{1}+R_{2}}\left[\begin{array}{ccc} 2 & 1 & 1\\ 0 & 1.5 & 0.5\\ 1 & 1 & 2 \end{array}\left|\begin{array}{c} 4\log2\\ 4\log3-2\log2\\ 8\log2 \end{array}\right.\right]$$

$$\overrightarrow{-\frac{1}{2}R_{1}+R_{3}}\left[\begin{array}{ccc} 2 & 1 & 1\\ 0 & 1.5 & 0.5\\ 0 & 0.5 & 1.5 \end{array}\left|\begin{array}{c} 4\log2\\ 4\log3-2\log2\\ 6\log2 \end{array}\right.\right]$$

$$\overrightarrow{R_{2}\leftrightarrow R_{3}}\left[\begin{array}{ccc} 2 & 1 & 1\\ 0 & 0.5 & 1.5\\ 0 & 1.5 & 0.5 \end{array}\left|\begin{array}{c} 4\log2\\ 6\log2\\ 4\log3-2\log2 \end{array}\right.\right]$$

$$\overrightarrow{-3R_{2}+R_{3}}\left[\begin{array}{ccc} 2 & 1 & 1\\ 0 & 0.5 & 1.5\\ 0 & 0 & -4 \end{array}\left|\begin{array}{c} 4\log2\\ 6\log2\\ 4\log3-20\log2 \end{array}\right.\right]$$

เปลี่ยนกลับมาเป็นสมการได้

$$\begin{eqnarray*} \log z & = & 5\log2-\log3\\ \log y & = & 3\log3-3\log2\\ \log x & = & \log2-\log3 \end{eqnarray*}$$

ดังนั้น $x=\frac{2}{3}$, $y=\frac{27}{8}$, $z=\frac{32}{3}$ และ

$$6a+8b-3c=4+27-32=-1$$