A+ » VCE » Maths Methods U3 & 4 Master Notes » A2 Algebra » Simultaneous Linear Equations

# Simultaneous Linear Equations

## 7.2 Simultaneous Linear Equations of More Variables

#### Simultaneous Linear Equations of Three Variables

• Consider a literal simultaneous linear equations with three variables:

\begin{cases}a_{1} x+b_{1} y+c_{1} z=d_{1} \\a_{2} x+b_{2} y+c_{2} z=d_{2} \\a_{3} x+b_{3} y+c_{3} z=d_{3}\end{cases}

The methods and skills involved are still the same as with two variables, namely elimination and substitution.

• It is also possible to solve such linear systems with the help of a calculator nowadays. Nevertheless, sometimes it might be quicker to solve it by hand if you realise variables can be cancelled out very quickly.
• To interpret it geometrically, the equation a x+b y+c z=d actually represents a plane in the 3-D space. Thus, solving the three simultaneous equations is actually same as finding the intersection point/line/plane of the 3 planes, if any.
Read More »7.2 Simultaneous Linear Equations of More Variables

## 7.1 Simultaneous Linear Equations of Two Variables

#### Geometry of Simultaneous Linear Equations

• Suppose you are asked to solve the simultaneous equations \begin{cases}x-y=2 \\2 x+3 y=10\end{cases}. In fact, each equation represents a line, if drawn on a 2-D coordinate plane.
• Thus, solving the simultaneous equations is similar to finding the intersection point of two lines.
• Consider solving two simultaneous equations for two unknowns. Interpreting geometrically, we want to find the intersection point(s) of two lines. There are three possible scenarios:
Read More »7.1 Simultaneous Linear Equations of Two Variables