IBDP Maths AHL 1.16 Solutions of systems of linear equations AA HL Paper 1- Exam Style Questions- New Syllabus
▶️ Answer/Explanation
Write the system as an augmented matrix.
\(\begin{bmatrix}1&0&-1&4\\2&2&1&14\\1&2&\alpha&\beta\end{bmatrix}\)
Apply the row operations \(R_2\rightarrow R_2-2R_1\) and \(R_3\rightarrow R_3-R_1\).
\(\begin{bmatrix}1&0&-1&4\\0&2&3&6\\0&2&\alpha+1&\beta-4\end{bmatrix}\)
Now apply \(R_3\rightarrow R_3-R_2\).
\(\begin{bmatrix}1&0&-1&4\\0&2&3&6\\0&0&\alpha-2&\beta-10\end{bmatrix}\)
For the system to have infinitely many solutions, the final row must represent the identity \(0=0\). Therefore, both entries in the final row must be zero.
\(\alpha-2=0\)
\(\alpha=2\)
and
\(\beta-10=0\)
\(\beta=10\)
With these values, there are only two independent equations in three unknowns, leaving one free variable. Therefore, the system has infinitely many solutions.
✅ Answer: \(\alpha=2\) and \(\beta=10\)