GDC: Solving Equations (DP IB Maths: AI HL)

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Systems of Linear Equations

What are systems of linear equations?

  • A linear equation is an equation of the first order (degree 1)
    • It is usually written in the form ax + by + c = 0 where a, b, and c are constants
  • A system of linear equations is where two or more linear equations work together
    • Usually there will be two equations with the variables x and y
    • The variables x and y will satisfy all equations
    • They are usually known as simultaneous equations
    • Occasionally there may be three equations with the variables x, y and z
  • They can be complicated to solve but your GDC has a function allowing you to solve them
    • The question may say ‘using technology, solve…’
      • This means you do not need to show a method of solving the system of equations, you can use your GDC

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

  • 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
  • Enter the equations into your calculator as you see them written
  • Your GDC will display the values of x and (or x, y, and z)

How do I set up a system of linear equations?

  • Not all questions will have the equations written out for you
  • There will be two bits of information given about two variables
    • Look out for clues such as ‘assuming a linear relationship’
  • Choose to assign x to one of the given variables and y to the other
    • Or you can choose to use more meaningful variables if you prefer
    • Such as c for cats and d for dogs
  • Write your system of equations in the form

a x plus b y equals e
c x plus d y equals f

  • Use your GDC to solve the system of equations
  • This function on the GDC can also be used to find the points of intersection of two straight line graphs
    • You may wish to use the graphing section on your GDC to see the points of intersection

Examiner Tip

  • Be sure to write down what you are putting into your GDC
    • If you have had to set up the system of equations as well make sure you write them down clearly before typing into your GDC

Worked example

A theme park has set ticket prices for adults and children.  A group of three adults and nine children costs $153 and a group of five adults and eleven children costs $211.

i)
Set up a system of linear equations for the cost of adult and child tickets.
 

ai-sl-1-1-4-systems-of-linear-equationsa

ii)
Find the price of one adult and one child ticket.
 

ai-sl-1-1-4-systems-of-linear-equationsb

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Polynomial Equations

What is a polynomial equation?

  • A polynomial is an algebraic expression consisting of a finite number of terms, with non-negative integer indices only
    • It is in the form a x to the power of n plus b x to the power of n minus 1 end exponent plus c x to the power of n minus 2 end exponent plus..., n element of straight natural numbers
  • A polynomial equation is an equation where a polynomial is equal to zero
  • The number of solutions (roots or zeros) depend on the order of the polynomial equation
    • A polynomial equation of order two can have up to two solutions
    • A polynomial equation of order five can have up to five solutions
  • A polynomial equation of an odd degree will always have at least one solution
  • A polynomial equation of an even degree could have no solutions

How do I use my GDC to solve polynomial equations?

  • You should use your GDC’s graphing mode to look at the shape of the polynomial
    • You will be able to see the number of solutions
    • This will be the number of times the graph cuts through or touches the x-axis
    • When entering a function into the graphing section you may need to adjust your zoom settings to be able to see the full graph on your display
    • Whilst in this mode you can then choose to analyse the graph
    • This will give you the option to see the solutions of the polynomial equation
      • This may be written as the zeros (points where the graph meets the x-axis)
  • Your GDC will also have a function within the algebra menu to solve polynomial equations
    • You will need to enter the order (highest degree) of the polynomial
    • This is the highest power (or exponent) in the equation
    • Enter the equation into your calculator
    • Your GDC will display the solutions (roots) of the equation
      • Be aware that your GDC may either show all solutions or only the first solution, it is always worth plotting a graph of the function to check how many solutions there should be

Examiner Tip

  • Be sure to write down what you are putting into your GDC
  • If you are using a graphical method it is often a good idea to sketch the graph that your GDC display shows
    • Don't spend too much time on this, a very quick sketch is fine

Worked example

For the polynomial equation 2 x cubed minus 2 x squared minus 3 x plus 4 equals 0:

i)
Use your GDC’s graphing function to sketch the graph of y equals 2 x cubed minus 2 x squared minus 3 x plus 4 and determine the number of solutions to the polynomial equation.

 

ai-sl-1-1-4-polynomials-a

ii)
Use your GDC to find the solution(s) of the polynomial equation.

 

ai-sl-1-1-4-polynomials-b

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Amber

Author: Amber

Expertise: Maths

Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.