9 The simultaneous equations
$$\begin{aligned}
& 2 x - y = 1
& 3 x + k y = b
\end{aligned}$$
are represented by the matrix equation \(\mathbf { M } \binom { x } { y } = \binom { 1 } { b }\).
- Write down the matrix \(\mathbf { M }\).
- State the value of \(k\) for which \(\mathbf { M } ^ { - 1 }\) does not exist and find \(\mathbf { M } ^ { - 1 }\) in terms of \(k\) when \(\mathbf { M } ^ { - 1 }\) exists.
Use \(\mathbf { M } ^ { - 1 }\) to solve the simultaneous equations when \(k = 5\) and \(b = 21\).
- What can you say about the solutions of the equations when \(k = - \frac { 3 } { 2 }\) ?
- The two equations can be interpreted as representing two lines in the \(x - y\) plane. Describe the relationship between these two lines
(A) when \(k = 5\) and \(b = 21\),
(B) when \(k = - \frac { 3 } { 2 }\) and \(b = 1\),
(C) when \(k = - \frac { 3 } { 2 }\) and \(b = \frac { 3 } { 2 }\).
RECOGNISING ACHIEVEMENT