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GDC: Solving Equations (DP IB Maths: AI HL)
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
- The question may say ‘using technology, solve…’
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
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.
ii)
Find the price of one adult and one child ticket.
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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 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 :
i)
Use your GDC’s graphing function to sketch the graph of and determine the number of solutions to the polynomial equation.
ii)
Use your GDC to find the solution(s) of the polynomial equation.
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