IB DP Maths: AI HL复习笔记1.1.3 GDC: Solving 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

What is a polynomial equation?

• 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