Solving Equations using a GDC (DP IB Applications & Interpretation (AI)) : Revision Note

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 y (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

• 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

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 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