Solve \(A v=b\) where \(A=\left[\begin{matrix}1 & 1 & 1\\0 & 8 & 1\\0 & 0 & 9\end{matrix}\right],\ b=\left[\begin{matrix}27\\83\\99\end{matrix}\right]\) and v=(x,y,z)ᵀ.
Find z. Explain your reasoning. [2]
Find y. Explain your reasoning. [2]
Find x. Explain your reasoning. [2]
Solution
Hint
The augmented constants must undergo the same row operations as the coefficients. Back-substitute from the final equation.
Worked solution
(a)
\((9)z=99\implies z=11\)
Answer: \(11\).
Answer: \(11\)
(b)
\(8y+11=83\implies y=9\)
Answer: \(9\).
Answer: \(9\)
(c)
\(x=27-9-11=7\)
Answer: \(7\).
Answer: \(7\)
Maths4U indicative marking guidance
Maths4U indicative marking guidance, not an official mark scheme. Credit valid alternative reasoning and consequential working; do not award a reasoning mark for an unexplained final answer.
(a) · 2 marks
1 mark: Back-substitute from the final equation.
1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not divide by a parameter without checking its zero case.
(b) · 2 marks
1 mark: Use the second equation after finding z.
1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Consistency concerns both A and b.
(c) · 2 marks
1 mark: Substitute y and z into the first equation.
1 mark: Obtain the stated result with the required sign, units and domain.
Alternative methods: Any complete mathematically valid method with the stated assumptions and equivalent result is acceptable.
Common error: Do not divide by a parameter without checking its zero case.