Systems of Linear Equations

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Introduction to Systems of Linear Equations

A linear equation is an equation of the first order (degree 1)
- This means that the maximum degree of each term is 1
- These are examples of linear equations:
  - 2x + 3y = 5 & 5x – y = 10 + 5z
- These are examples of non-linear equations:
  - x² + 5x + 3 = 0 & 3x + 2xy – 5y = 0
  - The terms x² and xy have degree 2
A system of linear equations is where two or more linear equations involve the same variables
- These are also called simultaneous equations
If there are n variables then you will need at least n equations in order to solve it
- For your exam n will be 2 or 3
A 2×2 system of linear equations can be written as
- $a_{1} x + b_{1} y = c_{1} a_{2} x + b_{2} y = c_{2}$
A 3×3 system of linear equations can be written as
- $a_{1} x + b_{1} y + c_{1} z = d_{1} a_{2} x + b_{2} y + c_{2} z = d_{2} a_{3} x + b_{3} y + c_{3} z = d_{3}$

The most common application of systems of linear equations is in geometry
For a 2×2 system
- Each equation will represent a straight line in 2D
- The solution (if it exists and is unique) will correspond to the coordinates of the point where the two lines intersect
For a 3×3 system
- Each equation will represent a plane in 3D
- The solution (if it exists and is unique) will correspond to the coordinates of the point where the three planes intersect

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Not all questions will have the equations written out for you
There will be bits of information given about the variables
- Two bits of information for a 2×2 system
- Three bits of information for a 3×3 system
- Look out for clues such as ‘assuming a linear relationship’
Choose to assign x, y & z to the given variables
- This will be helpful if using a GDC to solve
Or you can choose to use more meaningful variables if you prefer
- Such as c for the number of cats and d for the number of dogs

You can use your GDC to solve the system on the calculator papers (paper 2 & paper 3)
Your GDC will have a function within the algebra menu to solve a system of linear equations
You will need to choose the number of equations
- For two equations the variables will be x and y
- For three equations the variables will be x, y and z
If required, write the equations in the given form
- ax + by = c
- ax + by + cz = d
Your GDC will display the values of x and y (or x, y, and z)

Make sure that you are familiar with how to use your GDC to solve a system of linear equations because even if you are asked to use an algebraic method and show your working, you can use your GDC to check your final answer
If a systems of linear equations question is asked on a non-calculator paper, make sure you check your final answer by inputting the values into all original equations to ensure that they satisfy the equations

On a mobile phone game, a player can purchase one of three power-ups (fire, ice, electricity) using their points.

Adam buys 5 fire, 3 ice and 2 electricity power-ups costing a total of 1275 points.
Alice buys 2 fire, 1 ice and 7 electricity power-ups costing a total of 1795 points.
Alex buys 1 fire and 1 ice power-ups which in total costs 5 points less than a single electricity power up.

Find the cost of each power-up.

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