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

Determinant of the Third Order • Cramer’s Rule • Solving Systems of Equations by Determinants • Determinants on the Calculator

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}{rcl} a_{1} x + b_{1} y + c_{1} z = d_{1} \\ a_{2} x + b_{2} y + c_{2} z = d_{2} \\ a_{3} x + b_{3} y + c_{3} z = 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 = \frac{d_{1} b_{2} c_{2} + d_{3} b_{1} c_{2} + d_{2} b_{3} c_{1} - d_{3} b_{2} c_{1} - d_{1} b_{3} c_{2} - d_{2} b_{1} c_{3}}{a_{1} b_{2} c_{3} + a_{3} b_{1} c_{2} + a_{2} b_{3} c_{1} - a_{3} b_{2} c_{1} - a_{1} b_{3} c_{2} - a_{2} b_{1} c_{3}} \\ y = \frac{a_{1} d_{2} c_{3} + a_{3} d_{1} c_{2} + a_{2} d_{3} c_{1} - a_{3} d_{2} c_{1} - a_{1} d_{3} c_{2} - a_{2} d_{1} c_{3}}{a_{1} b_{2} c_{3} +} \end{array}

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Basic Technical Mathematics, 11th Edition by Allyn J. Washington, Richard Evans