1.10.1 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 work together
- 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

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.

1-10-1-ib-aa-hl-system-of-linear-equations-we-solution

DP IB Maths: AA HL

1.10.1 Systems of Linear Equations