DP IB Maths: AA HL

Revision Notes

1.10.1 Systems of Linear Equations

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Introduction to Systems of Linear Equations

What are 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:
      • + 5x + 3 = 0 & 3x + 2xy – 5y = 0
      • The terms 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 subscript 1 x plus b subscript 1 y equals c subscript 1
a subscript 2 x plus b subscript 2 y equals c subscript 2
  • A 3×3 system of linear equations can be written as
    • a subscript 1 x plus b subscript 1 y plus c subscript 1 z equals d subscript 1
a subscript 2 x plus b subscript 2 y plus c subscript 2 z equals d subscript 2
a subscript 3 x plus b subscript 3 y plus c subscript 3 z equals d subscript 3

What do systems of linear equations represent?

  • 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

Systems of Linear Equations

How do I set up a system of linear equations?

  • 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

How do I use my GDC to solve a system of linear equations?

  • 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)

Exam Tip

  • 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

Worked example

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

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Dan

Author: Dan

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.