# 5.6 Solving Systems of Three Linear Equations in Three Unknowns by Determinants

Just as systems of two linear equations in two unknowns can be solved by determinants, so can systems of three linear equations in three unknowns. The system

$$\begin{array}{r}{a}_{1}x\hspace{0.17em}+\hspace{0.17em}{b}_{1}y\hspace{0.17em}+\hspace{0.17em}{c}_{1}z\hspace{0.17em}=\hspace{0.17em}{d}_{1}\\ {a}_{2}x\hspace{0.17em}+\hspace{0.17em}{b}_{2}y\hspace{0.17em}+\hspace{0.17em}{c}_{2}z\hspace{0.17em}=\hspace{0.17em}{d}_{2}\\ {a}_{3}x\hspace{0.17em}+\hspace{0.17em}{b}_{3}y\hspace{0.17em}+\hspace{0.17em}{c}_{3}z\hspace{0.17em}=\hspace{0.17em}{d}_{3}\end{array}$$ **(5.10)**

can be solved in general terms by the method of elimination by addition or subtraction. This leads to the following solutions for `x`, `y`, and `z`.

$$\begin{array}{l}x\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{{d}_{1}{b}_{2}{c}_{2}\hspace{0.17em}+\hspace{0.17em}{d}_{3}{b}_{1}{c}_{2}\hspace{0.17em}+\hspace{0.17em}{d}_{2}{b}_{3}{c}_{1}\hspace{0.17em}-\hspace{0.17em}{d}_{3}{b}_{2}{c}_{1}\hspace{0.17em}-\hspace{0.17em}{d}_{1}{b}_{3}{c}_{2}\hspace{0.17em}-\hspace{0.17em}{d}_{2}{b}_{1}{c}_{3}}{{a}_{1}{b}_{2}{c}_{3}\hspace{0.17em}+\hspace{0.17em}{a}_{3}{b}_{1}{c}_{2}\hspace{0.17em}+\hspace{0.17em}{a}_{2}{b}_{3}{c}_{1}\hspace{0.17em}-\hspace{0.17em}{a}_{3}{b}_{2}{c}_{1}\hspace{0.17em}-\hspace{0.17em}{a}_{1}{b}_{3}{c}_{2}\hspace{0.17em}-\hspace{0.17em}{a}_{2}{b}_{1}{c}_{3}}}\\ y\hspace{0.17em}=\hspace{0.17em}{\displaystyle \frac{{a}_{1}{d}_{2}{c}_{3}\hspace{0.17em}+\hspace{0.17em}{a}_{3}{d}_{1}{c}_{2}\hspace{0.17em}+\hspace{0.17em}{a}_{2}{d}_{3}{c}_{1}\hspace{0.17em}-\hspace{0.17em}{a}_{3}{d}_{2}{c}_{1}\hspace{0.17em}-\hspace{0.17em}{a}_{1}{d}_{3}{c}_{2}\hspace{0.17em}-\hspace{0.17em}{a}_{2}{d}_{1}{c}_{3}}{{a}_{1}{b}_{2}{c}_{3}\hspace{0.17em}+\hspace{0.17em}}}\end{array}$$