IB Maths AA HL Ahl 1 16 Hl Systems of Linear Equations Questions

Practise solving systems of up to three linear equations, classifying solution conditions, writing parametric solutions and verifying the result.

Exam points

Solve systems of linear equations using elimination, substitution or matrix methods.
Interpret consistency and determine unknown variables or parameters using complete evidence.

[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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IB Maths AA HL Ahl 1 16 Hl Systems of Linear Equations Questions question 1