Solve the simultaneous equations
\begin{align}
4x - 2y + 3z &= 8,
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METHOD 1: \(\begin{pmatrix} 4 & -2 & 3 \\ 2 & -3 & 8 \\ 2 & 4 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \\ 0 \end{pmatrix}\) M1
Attempt to reduce to echelon form (formal matrix notation not required)
By row operations: eg. \(\begin{pmatrix} 1 & -3/2 & 4 \\ 0 & 4 & -13 \\ 0 & -1 & 17 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -1/2 \\ 10 \\ -19 \end{pmatrix}\) A1
Reduction to \(\begin{pmatrix} k & & \\ 0 & & \\ 0 & & \end{pmatrix}\)
Then \(\begin{pmatrix} 1 & -3/2 & 4 \\ 0 & -1 & 17 \\ 0 & 0 & 55 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -1/2 \\ -19 \\ -66 \end{pmatrix}\) A1
Reduction to \(\begin{pmatrix} k & & \\ 0 & & \\ 0 & 0 & \end{pmatrix}\)
Solving: \(z = -\frac{6}{5}\), \(y = -\frac{7}{5}\), \(x = \frac{11}{5}\) m1, A1
cao
[5] M0 for answers without supporting working
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| **METHOD 1:** | | |
| $\begin{pmatrix} 4 & -2 & 3 \\ 2 & -3 & 8 \\ 2 & 4 & -1 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} 8 \\ -1 \\ 0 \end{pmatrix}$ | M1 | Attempt to reduce to echelon form (formal matrix notation not required) |
| By row operations: eg. $\begin{pmatrix} 1 & -3/2 & 4 \\ 0 & 4 & -13 \\ 0 & -1 & 17 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -1/2 \\ 10 \\ -19 \end{pmatrix}$ | A1 | Reduction to $\begin{pmatrix} k & & \\ 0 & & \\ 0 & & \end{pmatrix}$ |
| Then $\begin{pmatrix} 1 & -3/2 & 4 \\ 0 & -1 & 17 \\ 0 & 0 & 55 \end{pmatrix} \begin{pmatrix} x \\ y \\ z \end{pmatrix} = \begin{pmatrix} -1/2 \\ -19 \\ -66 \end{pmatrix}$ | A1 | Reduction to $\begin{pmatrix} k & & \\ 0 & & \\ 0 & 0 & \end{pmatrix}$ |
| Solving: $z = -\frac{6}{5}$, $y = -\frac{7}{5}$, $x = \frac{11}{5}$ | m1, A1 | cao |
| | [5] | M0 for answers without supporting working |
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Solve the simultaneous equations
\begin{align}
4x - 2y + 3z &= 8, \\
2x - 3y + 8z &= -1, \\
2x + 4y - z &= 0.
\end{align} [5]
\hfill \mbox{\textit{WJEC Further Unit 4 2023 Q4 [5]}}