Q.Verify, by substitution, that $x=2,y=-1$ is indeed the solution of the system $3x+2y=4$, $x-3y=5$ (solved by Cramer's Rule in an earlier Worked Example), and explain why $D\neq0$ guarantees this is the ONLY solution.

Substituting the found values back into both original equations confirms the solution; a nonzero D means the two equations represent two genuinely independent (non-parallel) lines, which meet at exactly one point.

Substituting into the first equation

which matches the right-hand side, $4$. ✓

Substituting into the second equation

which matches the right-hand side, $5$. ✓

Both original equations are satisfied, so $x=2,y=-1$ is confirmed to be A solution.

Why $D\neq0$ guarantees it is the ONLY solution

Each linear equation in two unknowns represents a straight line on a graph. Two such lines can relate to each other in exactly three ways: they can cross at exactly ONE point (independent equations), never meet at all (parallel, inconsistent equations), or coincide completely (dependent equations, infinitely many common points). The coefficient determinant $D$ is exactly the quantity that distinguishes these cases: $D\neq0$ corresponds to the two lines having genuinely different slopes, so they cross at exactly one point — the single $(x,y)$ pair Cramer's Rule computes. Here $D=\begin{vmatrix}3&2\\1&-3\end{vmatrix}=-11\neq0$, so the lines $3x+2y=4$ and $x-3y=5$ a …

Treating the substitution check as sufficient on its own without understanding WHY $D\neq0$ matters — checking $D\neq0$ is what confirms uniqueness (ruling out the dependent case, where infinitely m …