Systems of Linear Equations (DP IB Analysis & Approaches (AA)): Revision Note

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 3×3 system of linear equations can be written as

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)