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Simultaneous 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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6.4 Systems with a Single Parameter

System of Equations with a Single Parameter

  • We have seen how simultaneous equations look like. For instance,

\left\{\begin{array}{ll}6=x+y+z & (1) \\ 4=4 x+2 y+z & (2)\end{array}\right.

and how there can be unique, infinite or no solutions for each set of simultaneous equations. These are explored in section “Solving Simultaneous Linear Equations” in A2 – Algebra.

  • Literal equations are very similar, just that instead of just numbers, we will have unknown constants. Such as,

\left\{\begin{array}{ll}x+y=2 m+3 & (1) \\ 3 x-2 y=7 m-1 & (2)\end{array}\right.

where m is a constant. Hence the solution (x,\ y) will be expressed in terms of m, and again there are unique, infinite, or no solutions. This will depend on the value(s) of m.

  • Parameters are a little different from literals. They are technically speaking, variables and not constants. It can be seen as a link between two (or more) variables.
  • For instance, take a cuboid with side length x for example. We denote the surface area as S, volume as V, then we have
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5.2 Simultaneous Equations: Polynomials

Solving Equations

  • Every now and then, you will be challenged with questions to solve .
  • Suppose we have y=f(x) and y=g(x) as two separate functions. If f(x)=g(x), it means that the graph of both functions intersect. Note that this tells nothing about the number of possible values for x.

Simultaneous Equations: Polynomials

  • We will first look at how this type of questions looks like, with polynomials first.
  • The steps are rather simple:
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