IB Maths AA HL 1.16 Systems of linear equations Question Bank

Practise IB Mathematics HL 1.16 by applying systems of linear equations methods to exam-style questions.

Exam points

identify the mathematical structure, variable or representation
select and apply the correct theorem, formula or algorithm
check the result using units, domain, graph or logical reasoning

[Maximum number: 5]

Find the values of k for which the following system of equations has no solutions and the value of k for the system to have an infinite number of solutions.

\begin{aligned} & x-3 y+z=3 \\ & x+5 y-2 z=1 \\ & 16 y-6 z=k \end{aligned}

The augmented matrix is
$\left(\begin{array}{ccc|c}1&-3&1&3\\1&5&-2&1\\0&16&-6&k\end{array}\right).$
Row operations (M1) give
$\left(\begin{array}{ccc|c}1&-3&1&3\\0&8&-3&-2\\0&0&0&k+4\end{array}\right).\quad (A1)(A1)$
Thus there are infinitely many solutions when $\boxed{k=-4}$ , and no solution when $\boxed{k\ne-4}$ . (R1)(R1)
Note: Other valid approaches are accepted.

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AHL 1.16 (HL)—Systems of linear equations question 1