Challenging +1.2 This is a standard Further Maths question on systems of equations requiring determinant calculation to show uniqueness, then finding consistency conditions when the determinant is zero. It involves routine matrix techniques (computing a 3×3 determinant, row reduction or substitution) with multiple steps, but follows a well-established template that FM students practice extensively. The conceptual demand is moderate—understanding when systems have unique vs infinite solutions—but the execution is algorithmic.
5 Show that if \(a \neq 3\) then the system of equations
$$\begin{aligned}
& 2 x + 3 y + 4 z = - 5 \\
& 4 x + 5 y - z = 5 a + 15 \\
& 6 x + 8 y + a z = b - 2 a + 21
\end{aligned}$$
has a unique solution.
Given that \(a = 3\), find the value of \(b\) for which the equations are consistent.
5 Show that if $a \neq 3$ then the system of equations
$$\begin{aligned}
& 2 x + 3 y + 4 z = - 5 \\
& 4 x + 5 y - z = 5 a + 15 \\
& 6 x + 8 y + a z = b - 2 a + 21
\end{aligned}$$
has a unique solution.
Given that $a = 3$, find the value of $b$ for which the equations are consistent.
\hfill \mbox{\textit{CAIE FP1 2006 Q5 [6]}}