Embedded Files
Welcome!
Welcome!
Here you will learn about systems of linear equations. You may have looked at these before in Year 11 as part of Algebra, which were then called simultaneous equations. Then, you had to solve a problem with 2 unknown variables, given 2 equations worth of information (2 x 2 system). In this topic we extend this to 3 unknown variables, given 3 equations (3 x 3 system) and start to look at Algebra in 3 dimensions! In addition, you will look at 3 x 3 systems that have either no solutions or infinitely many and what they might look like both contextually and as a 3D graph. The word solution here just means a point on a plane. A solution to a system of many planes is a point that lies on all planes.
This page has worksheets, videos, explanations and a practice assessment that cover the entire topic. When you have completed all the exercises you are ready for the assessment.
This video introduces the topic by first reviewing a 2 x 2 system and how it can be extended to a 3 x 3 system in 3 dimensions.
You can watch Mr Bradly's playlist of videos for this topic here:
Achieved level practice
Achieved level practice
Watch the video on solving a 3 x 3 system of equations, then complete the worksheets below. You can use the equation solver below to solve 3 x 3 systems quickly.
In this video Mr Davis uses an app called GeoGebra to view the system in 3D. You might want to try this too, however it is not permitted in your assessment for Merit and Excellence.
Achieved level practice (Google Site).pdfSystems Achieved practice.pdfEquation solver
Equation solver
You can use a device in this topic to solve systems of equations. We recommend you use this equation solver in Chrome. Click the image to get started.
Before you start each equation will need to be in the form:
ax + by + cz = d
This might require sometimes rearranging an equation. You then input the values a, b, c and d into the solver for all 3 equations and it will give you the unique solution for x, y and z.
Merit level practice
Merit level practice
So far you have looked at only unique solution systems. This means that there is only one combination of (x,y,z) that solves the problem. Graphically, this means that all three planes intersect each other at a single point. Solving and interpreting a unique solution system graphically or in context is an Achieved skill. To get Merit you will need to identify and interpret a non-unique system, such as the following:
A system will have infinite solutions if all three planes intersect along a single line. Unique and infinite solution systems are both called consistent because the solutions work for each equation.
A system will have no solution if either none of the planes intersect, or only 2 (but not all 3) intersect. This system is called inconsistent because a solution for one equation does not work for all three
Unique solution - point intersection
Unique solution - point intersection
Infinite solutions - line intersection
Infinite solutions - line intersection
No solution - 3 parallel planes
No solution - 3 parallel planes
No solution - 2 parallel planes
No solution - 2 parallel planes
No solution - triangular prism
No solution - triangular prism
After watching the above videos on the different types of systems, you should have a go at this activity.
For each scenario, form a 3 x 3 system of equations and either solve it or identify which non-unique solution system it is.
You will need to do this by looking for patterns or relationships within the equations and interpreting the system type geometrically. You can use GeoGebra to help you get started, but you will not be able to use it in the assessment.
The answers are at the end of the worksheet, so please mark your work before moving on.
Merit practice.pdfExcellence level practice
Excellence level practice
At Excellence level you will be asked to generalise and show deep understanding of each of the systems. Below are some examples of what this could look like:
You may be asked to describe what the equations would need to look like to give you a triangular prism type system (for example). What relationship needs to exist between the three equations?
You may need to change one equation to turn a system with no solutions into one with infinitely many!
You may also need to describe the relationship between the infinitely many solutions along a line.
The first two examples are answered in previous videos, however the last is covered in this video and involves a little algebra. After watching the video take any infinite solution system from previous problems (or make your own!) and use this method to find more than one solution to the problem.
Practice assessments
Practice assessments
Click below to generate a practice assessment. There are solutions at the end but they don't show working. Send your work to your teacher for feedback, then you are ready for the real assessment.
Page updated
Report abuse