Modelling with Simultaneous Equations
Learn modelling with simultaneous equations for NSW Year 12 Mathematics Standard 2. In this topic you turn a real-world story — two pricing plans, a mixture, a two-item purchase or a hire cost — into two linear equations, then solve them to find both unknowns.
You will learn to define the pronumerals, translate each fact into an equation, solve the pair by elimination, substitution or graphing, and interpret the solution in context with the right units — a core Standard 2 skill for solving practical problems.
Theory
Modelling with simultaneous equations turns a worded practical situation into two linear equations you can solve together. This Year 12 Standard 2 (NSW) guide shows how to define the pronumerals, translate each fact into an equation, solve by elimination, substitution or graphing, and interpret the answer in context — for pricing plans, mixtures, two-item purchases and hire costs.
Modelling with simultaneous equations means taking a real-world story with two unknown quantities and writing two linear equations that describe it, then solving them together to find both unknowns.
The key idea is that two unknowns need two facts. Each separate fact in the problem becomes one equation; a single equation on its own can never fix both values. Once you have a pair, you can solve it by elimination, substitution, or by graphing both lines and reading the point of intersection.
The final step is interpretation: the solution \((x,y)\) must be translated back into the context — a price, a mass, a number of hours — stated with the correct units. This is a core Year 12 Standard 2 (NSW) skill for pricing plans, mixtures, two-item purchases and hire costs.
A model with two unknowns is a pair of linear equations:
When two options are compared (two plans, two shops), model each as a cost and set the expressions equal to find where they agree:
How to model and solve a practical problem
- Define the pronumerals: state exactly what each letter represents, with its unit (dollars, kg, hours).
- Translate each fact in the story into one linear equation — you need two equations for two unknowns.
- Solve the pair by elimination, substitution, or by graphing both lines and reading the intersection.
- Interpret and check: write the answer in context with units, and verify it satisfies both original equations.
Let \(r\) and \(j\) be the prices (\$); eliminate \(r\) by doubling the second equation.
| \(4r+3j\) | \(=\) | \(16.50\) |
| \(2r+5j\) | \(=\) | \(13.50\) |
| \(4r+10j\) | \(=\) | \(27.00\) |
| \(7j\) | \(=\) | \(10.50\) |
| \(j\) | \(=\) | \(1.50\) |
| \(r\) | \(=\) | \(3.00\) |
A roll costs \(\$3.00\) and a juice \(\$1.50\).
Let \(p\) and \(s\) be the masses (kg); substitute \(s=10-p\).
| \(p+s\) | \(=\) | \(10\) |
| \(25p+15s\) | \(=\) | \(210\) |
| \(25p+15(10-p)\) | \(=\) | \(210\) |
| \(10p+150\) | \(=\) | \(210\) |
| \(p\) | \(=\) | \(6\) |
| \(s\) | \(=\) | \(4\) |
Use \(6\) kg premium and \(4\) kg standard beans.
Model each shop as a cost line, then set the two costs equal.
| \(25+10h\) | \(=\) | \(40+5h\) |
| \(5h\) | \(=\) | \(15\) |
| \(h\) | \(=\) | \(3\) |
| \(C\) | \(=\) | \(\$55\) |
Both cost \(\$55\) at \(3\) hours. For under \(3\) h Shop A is cheaper; beyond \(3\) h Shop B is cheaper.
Let \(x\) and \(y\) be the numbers of shots; subtract the equations to eliminate \(x\).
| \(x+y\) | \(=\) | \(15\) |
| \(x+2y\) | \(=\) | \(22\) |
| \(y\) | \(=\) | \(7\) |
| \(x\) | \(=\) | \(8\) |
She made \(8\) one-point and \(7\) two-point shots.
Common pitfalls
Frequently asked questions
How do you turn a worded problem into simultaneous equations?
Read the problem for the two unknown quantities and give each a pronumeral with its unit. Then translate each separate fact in the story into one linear equation. Two unknowns need two equations, so look for two distinct pieces of information.
Should I use substitution or elimination?
Both give the same answer, so use whichever is easier for the equations you have. Substitution is neat when one equation already has a pronumeral by itself, like s equals 10 minus p. Elimination is neat when you can scale the equations so one pronumeral cancels when you add or subtract.
Why do I have to define the pronumerals first?
Because the final answer must be stated in context. If you write let x and y with no meaning, you cannot say whether the solution is a price, a mass or a number of hours. Always write, for example, let r be the price of a roll in dollars.
Can I solve a modelling problem by graphing instead?
Yes. Model each condition as a straight line, graph both on the same axes, and read the coordinates of the point where they cross. This is especially useful when comparing two plans or hire costs, because the graph also shows which option is cheaper on each side of the crossing point.
What if the solution is not a whole number?
That is fine — prices and masses are often decimals, like a juice costing 1.50 dollars. Just make sure the answer makes sense in context. If a count of people or objects comes out as a fraction, re-check your equations, because those must be whole numbers.