4.03s Consistent/inconsistent: systems of equations

In the following set of simultaneous equations, \(a\) and \(b\) are constants. \begin{align} 3x + 2y - z &= 5
2x - 4y + 7z &= 60
ax + 20y - 25z &= b \end{align}

1. In the case where \(a = 10\), solve the simultaneous equations, giving your solution in terms of \(b\). [3]
2. Determine the value of \(a\) for which there is no unique solution for \(x\), \(y\) and \(z\). [3]
3. 1. Find the values of \(\alpha\) and \(\beta\) for which \(\alpha(2y - z) + \beta(-4y + 7z) = 20y - 25z\) for any \(y\) and \(z\). [3]
   2. Hence, for the case where there is no unique solution for \(x\), \(y\) and \(z\), determine the value of \(b\) for which there is an infinite number of solutions. [2]
   3. When \(a\) takes the value in part (ii) and \(b\) takes the value in part (iii)(b) describe the geometrical arrangement of the planes represented by the three equations. [1]

The matrix \(\mathbf{A}\) is given by \(\mathbf{A} = \begin{pmatrix} a & 2 & 3 \\ 4 & 4 & 6 \\ -2 & 2 & 9 \end{pmatrix}\) where \(a\) is a constant. It is given that if \(\mathbf{A}\) is not singular then $$\mathbf{A}^{-1} = \frac{1}{24a-48} \begin{pmatrix} 24 & -12 & 0 \\ -48 & 9a+6 & 12-6a \\ 16 & -2a-4 & 4a-8 \end{pmatrix}.$$

1. Use \(\mathbf{A}^{-1}\) to solve the simultaneous equations below, giving your answer in terms of \(k\). \begin{align} x + 2y + 3z &= 6
   4x + 4y + 6z &= 8
   -2x + 2y + 9z &= k \end{align} [3]
2. Consider the equations below where \(a\) takes the value which makes \(\mathbf{A}\) singular. \begin{align} ax + 2y + 3z &= b
   4x + 4y + 6z &= 10
   -2x + 2y + 9z &= -13 \end{align} \(b\) takes the value for which the equations have an infinite number of solutions.
3. For the equations in part (ii) with the values of \(a\) and \(b\) found in part (ii) describe fully the geometrical arrangement of the planes represented by the equations. [2]