Three Variable Systems

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📚 Understanding 3x3 Systems

WALT (We Are Learning To): Understand the structure of 3x3 systems and translate word problems into three-variable equations.

What is a 3x3 System?
A system with three equations and three variables (usually x, y, and z). Each equation shows a relationship between all three variables.

Example:
x + y + z = 10
2x - y + 3z = 15
x + 2y - z = 5

1. How many equations do you need to solve for three unknown variables?

Two equations

Three equations

Four equations

2. In the system above, what are the three variables?

Variables: _____, _____, and _____

✏️ Translating Word Problems

3. Cinema Tickets Problem:
At the local cinema, adult tickets cost $15, student tickets cost $10, and child tickets cost $8. On Saturday, they sold 120 tickets and collected $1350. There were twice as many adult tickets sold as child tickets.

Let a = adult tickets, s = student tickets, c = child tickets

Write the three equations:

Total tickets: a + _____ + _____ = _____

Total money: _____a + _____s + _____c = _____

Adult-child relationship: _____ = _____

4. School Fundraiser:
The school sold pies ($5), sausage rolls ($3), and drinks ($2). They sold 200 items, made $650, and sold 50 more pies than drinks.

Define your variables and write the system:

🔢 Solving Simple Systems

5. Solve this system using substitution or elimination:

x + y + z = 6
x - y = 0
y + z = 4

Solution: x = _____, y = _____, z = _____

6. Extension Challenge - Waiheke Ferry Problem:
Three ferry routes operate from Waiheke Island. Route A takes 35 minutes, Route B takes 40 minutes, and Route C takes 45 minutes. In one day, 18 ferries ran, taking a total of 705 minutes. Route A ran 3 more times than Route C.

Set up and solve the system to find how many times each route operated: