Using row reduction, find the solutions in terms of \(a\) and \(b\) when \(a\) ≠ 3 .

Explain why the equations have no unique solution when \(a\) = 3.

Find all the solutions to the equations when \(a\) = 3, \(b\) = 10 in the form r = s + \(\lambda \)t.

\(\left( {\begin{array}{*{20}{c}}
1 \\
3 \\
2

\end{array}\,\,\begin{array}{*{20}{c}}

0 \\ 

1 \\
a

\end{array}\,\,\begin{array}{*{20}{c}}

1 \\

2 \\

{-1}

\end{array}\,\,\begin{array}{*{20}{c}}
{ - 1} \\
1 \\
b
\end{array}} \right) \Rightarrow \left( {\begin{array}{*{20}{c}}
1 \\
{3a - 2} \\
2
\end{array}\,\,\begin{array}{*{20}{c}}

0 \\
0 \\

a
\end{array}\,\,\begin{array}{*{20}{c}}

1 \\

{2a+1} \\

{-1}

\end{array}\,\,\begin{array}{*{20}{c}}
{ - 1} \\
{a - b} \\
b
\end{array}} \right)\) or equivalent        M1A1

\(\left( {\begin{array}{*{20}{c}}
1 \\
{3a - 2} \\
2
\end{array}\,\,\begin{array}{*{20}{c}}
0 \\
0 \\
a
\end{array}\,\,\begin{array}{*{20}{c}}
{a - 3} \\
{2a + 1} \\
{ - 1}
\end{array}\,\,\begin{array}{*{20}{c}}
{ - 4a + 2 + b} \\
{a - b} \\
b
\end{array}} \right)\)

\(z = \frac{{ - 4a + b + 2}}{{a - 3}}\)      M1A1

\(x =  - 1 - z\)      M1

\(x =  - 1 - \left( {\frac{{ - 4a + b + 2}}{{a - 3}}} \right)\)

\(x = \frac{{ - a + 3 + 4a - b - 2}}{{a - 3}}\)

\(x = \frac{{3a - b + 1}}{{a - 3}}\)      A1

\(y = 1 - 3x - 2z\)       M1

\(y = 1 - 3\left( {\frac{{3a - b + 1}}{{a - 3}}} \right) - 2\left( {\frac{{ - 4a + b + 2}}{{a - 3}}} \right)\)

\( = \frac{{a - 3 - 9a + 3b - 3 + 8a - 2b - 4}}{{a - 3}}\)

\( = \frac{{b - 10}}{{a - 3}}\)     A1

[8 marks]

when \(a\) = 3 the denominator of \(x\), \(y\) and \(z\) = 0      R1

Note: Accept any valid reason.

hence no unique solutions       AG

[1 mark]

For example let \(z = \lambda \)      (M1)

\(x =  - 1 - \lambda \)      (A1)

\(y = 1 - 3\left( { - 1 - \lambda } \right) - 2\lambda \)

\(y = 4 + \lambda \)      (A1)

r = \(\left( \begin{gathered}
- 1 \hfill \\
4 \hfill \\
0 \hfill \\
\end{gathered} \right) + \lambda \left( \begin{gathered}
- 1 \hfill \\
1 \hfill \\
1 \hfill \\
\end{gathered} \right)\)      A1

[4 marks]

Note: Accept answers which let \(x = \lambda \) or \(y = \lambda \).