Q.Examine the consistency of the system $2x + 3y = 5$ and $4x + 6y = 10$, and state the number of solutions.

Consistency is decided from the determinants $D$, $D_x$, $D_y$: here all three turn out to be zero, which signals infinitely many solutions.

Compute the three determinants. $D = \begin{vmatrix} 2 & 3 \\ 4 & 6 \end{vmatrix} = (2)(6)-(3)(4) = 12-12 = 0.$ $D_x = \begin{vmatrix} 5 & 3 \\ 10 & 6 \end{vmatrix} = (5)(6)-(3)(10) = 30-30 = 0, \qquad D_y = \begin{vmatrix} 2 & 5 \\ 4 & 10 \end{vmatrix} = (2)(10)-(5)(4) = 20-20 = 0.$

Since $D = 0$ and $D_x = D_y = 0$, the system is consistent with infinitely many solutions — the two equations are dependent. …

One can also test consistency by comparing ratios: for a1 x + b1 y = c1 and a2 x + b2 y = c2, when a1/a2 = b1/b2 = c1/c2 (here 2/4 = 3/6 = 5/10 = 1/2) th …

A common error is to conclude 'no solution' as soon as D = 0. D = 0 alone is not enough - one must also check D_x and D_y: all zero means infinitely many solutions (consistent), …