Chapter 11 - Systems of Equations and Inequalities - 11.3 Systems of Linear Equations: Determinants - 11.3 Assess Your Understanding - Page 743: 54

$x=-\frac{7}{6}.$

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 $D=3(x\cdot(-2)-5\cdot1)-2(1\cdot(-2)-5\cdot0)+4(1\cdot1-x\cdot0)=0\\3(-2x-5)-2(-2)+4(1)=0\\-6x-15+4+4=0\\-6x=7\\x=-\frac{7}{6}.$