---
title: "Simultaneous equations Practice Questions & Worked Answers"
description: "Solve two equations at once by substitution or elimination, and recognise when there is no solution at all. Free Years 9 to 10 practice with worked solu..."
image: "https://nobleeducators.com/og-image.jpg"
url: "https://nobleeducators.com/australia/practice/simultaneous-equations"
---

Your Australian school pathway · Noble Educators

# Simultaneous equations practice questions and worked solutions

Simultaneous equations ask for the values that satisfy two conditions at the same time. Graphically the solution is where two lines cross, which is why a pair can have one solution, none, or infinitely many. Students usually learn substitution and elimination as separate procedures and then apply whichever they met most recently, rather than choosing based on the equations in front of them.

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2

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3

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## Choosing substitution or elimination

Substitution is faster when one variable is already isolated, or can be isolated in one step without introducing fractions. If an equation reads y = 2x + 1, substituting that expression directly into the other equation is almost always the shortest route to an answer.

Elimination is faster when the equations are both in the form ax + by = c, particularly when adding or subtracting them cancels a variable immediately. A pair such as x + y = 10 and x − y = 4 is built for elimination, and a student who sets up substitution instead has made more work than the question required.

Neither method is more correct, and marks are awarded for either. What examiners are looking for is a correct and clearly set-out approach, so spending three seconds deciding which is cleaner is a better investment than starting immediately with whichever is more familiar.

## Checking your answer properly

Substitute the values back into the equation you did not use when finding the second variable. Checking against the equation you just used will confirm your arithmetic in that step but cannot catch an error made earlier, so it gives false confidence in exactly the situation where a check matters.

For word problems, the check has a second part: confirm that the answer makes sense in context. A negative number of tickets or a fractional number of buses signals a set-up error rather than an arithmetic one, and the marks lost there are for translating the problem rather than for solving it.

## No solution and infinitely many

When the variables cancel and leave a false statement such as 0 = 6, the lines are parallel and distinct, so there is no solution. When they cancel and leave a true statement such as 0 = 0, the two equations describe the same line and every point on it is a solution.

These outcomes feel like something has gone wrong, and students often assume they have made a mistake and start over. Recognising the two signatures saves time and picks up marks, because questions asking you to determine the number of solutions are testing precisely this recognition rather than the algebra.

There is a parameter version of the same question at senior level, asking for the value of a constant that makes a system have no solution. Those are answered by setting the coefficients proportional to each other, which follows directly from the lines being parallel.

## Setting up word problems

Most marks in this topic are lost before any algebra happens. Define the variables explicitly in words, write the two relationships as separate sentences, and only then translate each sentence into an equation. Attempting to write the equations straight from the paragraph is where the translation errors creep in.

Watch for relationships expressed as differences and comparisons, which reverse easily. A statement that one number is four more than another becomes a minus sign on one side or a plus on the other depending on which variable you subtract from which, and getting it backwards produces a clean-looking answer that is wrong.

## Practise it now

5 questions

Choose an answer and the method appears straight away. Nothing is recorded and there is no time limit — work through them at whatever pace is useful, and read the working even on the ones you get right.

1. Question 1. Solve the pair x + y = 10 and x − y = 4.x = 7, y = 3x = 3, y = 7x = 6, y = 4x = 5, y = 5
2. Question 2. If 2x + y = 11 and x = 3, what is y?58174
3. Question 3. Solve 3x + 2y = 12 when x = 2.y = 3y = 6y = 4y = 9
4. Question 4. Two lines are graphed and turn out to be parallel but distinct. How many solutions does the pair of equations have?NoneOneTwoInfinitely many
5. Question 5. What happens when you try to solve y = 2x + 1 together with 2y = 4x + 2?There are infinitely many solutionsThere is exactly one solutionThere are no solutionsThe second equation is invalid

Years 9 to 10 · Australian Curriculum v9.0 content descriptors AC9M10A02, AC9M9A02. Descriptor codes are published by ACARA and are quoted so you can check this practice against your school’s scope and sequence. Noble Educators is an independent tutoring provider and is not affiliated with ACARA.

Common questions

## What families ask about this

### When are simultaneous equations taught in Australia?

Linear simultaneous equations are a Year 9 and Year 10 topic under the Australian Curriculum, and they carry into every senior mathematics course. Students meet non-linear versions, such as a line intersecting a parabola, in Year 11.

### Can a graphics calculator solve them for me?

In courses and assessment components that permit one, yes, and knowing how to use it is a legitimate skill. Non-calculator sections still require the algebraic method, and questions that ask you to show working award marks for the process rather than the final value.

### What if there are three equations and three unknowns?

The same logic extends: eliminate one variable to reduce the system to two equations in two unknowns, then solve as usual. This appears in senior specialist courses and in matrix methods, and it is the natural generalisation of elimination rather than a new technique.

### Why does my answer differ when I use the other method?

It should not — both methods must give the same solution, so a difference means an arithmetic error in one of them. Substituting your values into both original equations identifies which attempt is wrong, since the correct solution satisfies each equation exactly.

## What sessions cost

Every year level has a published per-session rate in AUD, with no lock-in contract.

Year 1$12

AUD per session

Year 7$18

AUD per session

Year 12$26

AUD per session

Booking a block reduces the per-session rate: 5% from 8 sessions, 8% from 12 sessions, 12% from 15 sessions, 15% from 20 sessions. At 12 sessions, Year 1 works out to $11.04 per session. The first lesson is free and there is no lock-in contract.

[See the full rate card](https://nobleeducators.com/australia#pricing)

## How that compares

Online 1-on-1, a private in-person tutor and a tutoring centre are genuinely different products, and the cheapest is not automatically the right one. These are the typical advertised rates for each in Australia.

| Option | Typical cost per session | Format |
| --- | --- | --- |
| Noble Educators | $12–$34 | 1-on-1, live |
| Private in-person tutor | $40–$100 | 1-on-1, in person |
| Tutoring centre | $45–$90 | Small group, sometimes 1-on-1 |
| Online marketplace tutor | $30–$70 | 1-on-1, live |

Ranges describe typical advertised rates for each type of tutoring in Australia and are provided as general guidance only. They are not quotes from specific providers, and actual prices vary by tutor, year level, subject, location and session length. Compare current published pricing directly before deciding.

[See the full cost comparison](https://nobleeducators.com/australia#compare)

For parents

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## What to share before the first lesson

The tutor match works from evidence, not a job title. Anything from the list below gives the first session a concrete starting point instead of a cold diagnostic.

- The most recent school report or assessment result
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- Any teacher feedback, and the topic where confidence broke

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