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

Amber

Written by: Amber

Reviewed by: Mark Curtis

Updated on

DP Applications & Interpretation (AI): HLDP Applications & Interpretation (AI): SL

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Systems of Linear Equations

What are systems of linear equations?

  • A system of linear equations is a set of n simultaneous equations in n unknowns (called an n cross times n system)

    • The word linear means there are no squared or higher power terms, and no cross terms (e.g. x y)

  • A 1 cross times 1 linear system has the form

a x plus b equals c

  • A 2 cross times 2 linear system has the form

table row cell a x plus b y end cell equals e row cell c x plus d y end cell equals f end table

  • A 3 cross times 3 linear system has the form

table row cell a x plus b y plus c z end cell equals p row cell d x plus e y plus f z end cell equals q row cell g x plus h y plus j z end cell equals r end table

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

  • Your GDC has a function within the algebra menu to solve a system of linear equations

    • Enter the equations into your calculator as you see them written

    • Your GDC will solve it and display the values of x and y (or x, y and z)

Examiner Tips and Tricks

If an exam question says 'using technology, solve...' then you can use your GDC to solve the equations and no method is needed.

How do I form a system of linear equations?

  • You may have to create your own system of linear equations from a given worded context

    • Assign appropriate letters to your variables (x, y and z are the default)

    • Read the context carefully and convert the words into equations

  • Then solve the system of linear equations using your GDC as above

Examiner Tips and Tricks

If you form your own equations, write out the system of equations clearly for the examiner before using your GDC to solve it.

How do I find the point of intersection of two straight lines?

  • Rearrange the equations of the two straight lines into the form

table row cell a x plus b y end cell equals e row cell c x plus d y end cell equals f end table

  • Then solve the system of equations using your GDC as above

    • Write your answers as coordinates, open parentheses x comma space y close parentheses

Examiner Tips and Tricks

If you need to see the point of intersection on a graph, you can use the graphing mode on your GDC.

Worked Example

A theme park has different 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.

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

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

(b) Using technology, find the price of one adult and the price of 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 equation has the form

a subscript n x to the power of n plus a subscript n minus 1 end subscript x to the power of n minus 1 end exponent plus a subscript n minus 2 end subscript x to the power of n minus 2 end exponent plus... plus a subscript 2 x squared plus a subscript 1 x plus a subscript 0 equals 0

  • n must be a positive integer and is called the order (or degree)

    • Examples include

      • 2 x cubed plus 4 x squared minus x plus 1 equals 0 (order 3)

      • x to the power of 6 plus 2 x to the power of 5 minus x to the power of 4 plus 4 x cubed plus x squared plus 10 x minus 4 equals 0 (order 6)

      • x to the power of 10 minus 1 equals 0 (order 10)

      • 2 x minus 3 equals 0 (order 1)

  • The following are not polynomials

    • x squared plus x to the power of negative 3 end exponent equals 0 (no negative powers allowed)

    • 4 x plus x to the power of 1 half end exponent minus 1 equals 0 (no non-integer powers allowed)

How do I use my GDC to solve polynomial equations?

  • Your GDC has a function within the algebra menu to solve polynomial equations

    • Enter the order (degree) of the polynomial

    • Enter the equation into your calculator

    • Your GDC will then display the solutions (roots) of the equation

Examiner Tips and Tricks

If there are lots of solutions to a polynomial equation, your GDC may only show the first few solutions (you will need to scroll along to find the others).

How many solutions does a polynomial equation have?

  • A polynomial equation of order n can have up to n solutions

    • If n is odd, there will always be at least one solution

    • If n is even, there could be no solutions

  • You can use your GDC’s graphing mode to find the number of solutions

    • Plot the equation y equals...

    • Count the number of times the graph cuts (or touches) the x-axis

      • You may need to adjust your zoom settings to see the full graph

Examiner Tips and Tricks

In graphing mode on your GDC, choosing to 'analyse' the graph then selecting 'zeros' gives you an alternative way to solve polynomial equations.

Worked Example

Consider the equation 2 x cubed minus 2 x squared minus 3 x plus 4 equals 0.

(a) Use technology to sketch the graph of y equals 2 x cubed minus 2 x squared minus 3 x plus 4 and hence determine the number of solutions to the equation.

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

(b) Use your GDC to find all the possible solution(s) to the equation.

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

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    Author
    Reviewer
    Amber

    Author: Amber

    Expertise: Maths Content Creator

    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.

    Mark Curtis

    Reviewer: Mark Curtis

    Expertise: Maths Content Creator

    Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.