# Equations and Expressions: Level 8

The key idea of equations and expressions at levels 7 and 8 is that equations can be treated as objects, which can undergo a variety of transformations while preserving the same solutions.

At Level 8 the new class of numbers introduced is that of complex numbers.  The number of quadratic equations that may be solved is again extended from those with real solutions to those with complex solutions.  The manipulation of complex numbers required is an extension of the manipulation of surds and hence the work on surds at Level 7 Equations and Expressions forms a foundation for the work on complex numbers.

The use of equations to solve problems in context takes on even more importance at Level 8.  Instead of using physical situations as a model for understanding the mathematics, the emphasis is on using mathematical models to predict properties of the situations.  Understanding the limitations of the mathematical model and the meaning (or otherwise) of mathematical solutions to real problems is therefore very important.

As students are expected to form and solve polynomial, hyperbolic and trigonometric equations the connections to Level 7 and Level 8 Patterns and Relationships become indispensible, as does the use of graphics calculators.  The domain of trigonometric equations is extended from [0, 2π] and manipulation of trigonometric expressions, making use of trigonometric identities, is required.  This manipulation of trigonometric expressions is also required for Level 8 Calculus.

The work on simultaneous equations in Level 7 Equations and Expressions  is extended to three linear equations in three unknowns.  This has applications to linear programming in Level 8 Patterns and Relationships.

All of this work has important applications and extensions in later work.  For examples, the trigonometry is useful in solving differential equations and as a modelling tool and for approximating functions.  The work on solving simultaneous equations extends to n equations and m unknowns.  Complex numbers are useful in solving modelling problems and can be applied to solve classical Euclidean geometry problems, while the theory behind calculus in real analysis is extendable to complex analysis.

This key idea develops from the key idea of equations and expressions at level 7 that that linear relationships between variables can be represented by a linear equation.