AS 3.4 Equation Solving

1

Topic overview & Achievement criteria
SIMULTANEOUS LINEAR EQUATIONS
with 2 variables. Review of level 2 simultaneous equations:
ax + by + c = 0
dx + ey + f = 0
One, none or infinite solutions and the graphical representation of these situations.
Solving 2 variable simultaneous equations using: Substitution, Elimination, Rearranging, and Scaling algebraic skills.

Class notes, Blank Notes, Examples

Ex 10.1 p182

Simultaneous Equations - 2 variables

parabola

2

Graphical representation of 3-Dimensional equations. 3 dimensional x,y,x cartesian exis sustem. Simple functions and their graphs. Sketching planes in 3d space using the intercept method.

Graph calc Demo Graph Calc download (Excellent 3D graphing freeware)

Download
Google sketchup download

3

Solving 3 Equations, Unique Solution. Solving by Substitution and by Elimination.

Method: 1) Select a pair of equations and eliminate one variable
2) Select another pair of equations and eliminate the same variable.
3) Solve the pair of 2-variable equations to find two solutions.
4) Substitute the two solutions back into an initial equation to find the third variable.

Class notes, Blank Notes

Ex 10.2 p190

3 variable simultaneous equations

4

Solving 3 Equations - lots of practice...

5

Possible Solutions

Determining and recognising when a system of equations has NO solution. Identifying graphically what is occurring.

	No Solution (inconsistent equations)

Determining and recognising when a system of equations has INFINITE solutions. Identifying graphically what is occurring.

	Infinite Solutions (dependent equations)

Class notes, Blank Notes

Ex 10.03A p195 | Ex 10.03B p195

Graphical representation of possible solutions

6

Solving application problems. Interpreting 'word' problems and forming a system of simultaneous equations before solving them.

Class notes, Blank Notes

Ex 10.04A p199 | Ex 10.04B p199

Takeaway examples

7

Linear programming is a method of determing the maximum or minimum of a function(eg profit) given a set of restrictions (constraints). This involves inequality functions and graphs with shaded regions.

Class notes, Blank Notes

Ex 12.01 p231 | Ex 12.02 p231

8

Practice of forming mathematical inequalities from long and horrible english sentences.

Class notes, Blank Notes

Ex 12.03 p233 | Ex 12.04A p235 | Ex 12.04B p235

Forming Inequalities - woldo maths

9

Moving line approach

Applied Linear programming: Sketching simple feasible regions and calculating maximum and minimum values within these constraints. The moving line approach Demo - (enable macros)

Class notes, Blank Notes

Ex 12.05 p239 | Ex 12.06A p243

Demo - (enable macros)

10

Linear programming practice

Class notes, Blank Notes

Ex 12.06B p243

11

Linear programming practice

More practice of linear programming.

Class notes, Blank Notes

Ex 12.06C p243

12

Numerical Solutions

Overview to numerical methods to solving equations in the form f(x)=0. Forming a table of values to determine where solutions occurr. Terminology: root of a function.

An overview of the Bisection-Method

Class notes, Blank Notes

Ex 11.01 p207

13

Bisection practice

Constructing a table to carry out several iterations of the bisection method. Determining the level of accuracy of a result after 'n' iterations

Class notes, Blank Notes

Ex 11.04 p214

14

Newton-Raphson method

An overview of the Newton-Raphson method and a reminder of basic differentiation. Using the method to solve equations using the calculator ANS function to allow fast iterations.

Class notes, Blank Notes

Ex 11.06 p221

Waldos: Newton-Raphson method

15

Rearranging Newton-Raphson

Rearranging the Newton-Raphson formula into a simpler form. Solving more problems with Newton-Raphson method   

Class notes, Blank Notes

Ex 11.08 p224 | Ex 11.09A p225

16

Comparison of each method

Using the Newton-Raphson formula with trig, log and exponential functions. Comparing the two methods in terms of speed and accuracy. When can the Newton-Raphson go wrong...

Class notes, Blank Notes

Ex 11.10 p226 | Ex 11.11 p227

17

Revision

Solving equations practice. Simultaneous equations, Linear programming, Numerical Methods  

Class notes, Blank Notes

Popes revision sheets
1 2 3 4 5 6 7 8 9 10

18

Objective: Practice Assessment for 3.4