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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.
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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:
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