Precalculus (10th Edition)

Published by Pearson
ISBN 10: 0-32197-907-9
ISBN 13: 978-0-32197-907-0

Chapter 11 - Systems of Equations and Inequalities - 11.3 Systems of Linear Equations: Determinants - 11.3 Assess Your Understanding - Page 743: 55


$x=0$ or $x=-9$.

Work Step by Step

We know that for a matrix \[ \left[\begin{array}{rrr} a & b & c \\ d &e & f \\ g &h & i \\ \end{array} \right] \] the determinant, $D=a(ei-fh)-b(di-fg)+c(dh-eg).$ Hence here $D=x(x\cdot(-2)-0\cdot1)-2(1\cdot(-2)-0\cdot6)+3(1\cdot1-x\cdot6)=0\\x(-2x)-2(-2)+3(1-6x)=7\\-2x^2+4+3-18x=7\\2x^2+18x=0\\2x(x+9)=0.$ Hence by the zero product property $x=0$ or $x+9=0\\x=-9$.
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