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Three Variable Systems

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Three Variable Systems

A three-variable linear system has three unknowns linked by equations. Show your algebraic working, state any restrictions, and check solutions in the original equations. Use exact values unless a question asks otherwise.

Structure and modelling

Read each question carefully: some ask for a model, and others ask you to solve or analyse one.

1.A three-variable system has unknowns x, y and z. Explain why one equation generally cannot determine all three values uniquely. What information do three independent equations provide?

2.A theatre sells adult, student and child tickets. Let a, s and c be the numbers sold. There are 180 tickets in total, adult tickets cost $24, student tickets cost $16 and child tickets cost $10, and the total takings are $3,720. There are 20 more student tickets than child tickets. Write a system of three equations to model the situation.

3.The ticket model gives three equations for a, s and c. Solve the system from the previous question and check that your ticket counts satisfy both the total number of tickets and total takings.

Solve and analyse

Choose an efficient elimination or substitution strategy. Keep track of how each equation changes.

4.Solve the system: (1) x + y + z = 6; (2) 2x - y + z = 3; (3) x + 2y - z = 2. Give the ordered triple (x, y, z) and verify it in all three equations.

5.A student eliminates variables from a system and obtains 0 = 5. What does this tell you about the original system? Choose the best conclusion.

The system has exactly one solution.
The system has no solution.
The system has infinitely many solutions.

6.Consider the system x + y + z = 1, x + 2y + 3z = 2, and 2x + 3y + λz = 4. Determine the value of λ for which the system is inconsistent. Explain your reasoning.

7.For the same system, now set λ = 5. Find the unique solution (x, y, z).

8.A system reduces to x + 2y - z = 4, y + 3z = 1, and 0 = 0. State whether it has no solution, exactly one solution, or infinitely many solutions. Then express x and y in terms of a free variable and give one solution.

9.A machine produces three products, A, B and C. In one shift it makes 30 items altogether. Product A takes 2 minutes per item, B takes 3 minutes and C takes 5 minutes; production uses 110 minutes. It makes twice as many A items as C items. Let a, b and c be the quantities. Write the system and determine whether it has a solution. Interpret your result in context.

10.A classmate claims that any system of three linear equations in three variables must have exactly one solution. Give a counterexample and identify whether your example has no solutions or infinitely many solutions.