Level 3 Complex Numbers Walkthrough

2025 NCEA Level 3 Complex Numbers Question 3(e)

2025 Paper

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Question

Given that \(u\) and \(v\) are both complex numbers, solve the following pair of simultaneous equations, giving solutions in the form \(a+bi\).

\[ ui + 2v = 3 \] \[ u + (1-i)v = 4 \]

First walkthrough idea

Focus to try first

using elimination with a strategically chosen multiple, then simplifying the resulting complex fractions to solve for \(u\) and \(v\).

Step 1

Make the \(u\)-terms match

That gives \(ui+(1-i)vi=4i\), so the \(ui\)-terms are ready to eliminate.

Show the first step’s working

That gives \(ui+(1-i)vi=4i\), so the \(ui\)-terms are ready to eliminate.

Key result

Multiply \(u+(1-i)v=4\) by \(i\).

Walkthrough overview

What this question practises

This 2025 walkthrough is part of AS91577 — Apply the algebra of complex numbers in solving problems.

Method: Solving simultaneous equations with complex numbers.

This is Question 3(e) from the 2025 NCEA Level 3 Complex Numbers paper for AS91577 — Apply the algebra of complex numbers in solving problems. Use the guided hints to practise the method before revealing the full working.

Calc.nz is an independent learning resource. Compare questions, diagrams, and assessment information with the official NZQA resources for AS91577.

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Learning summary

Review the method, not only the answer

This walkthrough helps you practise solving simultaneous equations with complex numbers. Use the hints to plan the method, then repeat the question without hints and check each step.

Common mistake to avoid

Check each step against the original condition, preserve signs and restrictions, and confirm that the final result answers the question asked.

Continue practising

← Question 3(d)