#### Chapter 4 System of Linear Equations

Section 4.3 LS in three Variables

# 4.3.1 Introduction

In the following section we will slightly increase the level of difficulty and discuss slightly more complex systems.
##### Example 4.3.1
While playing, three children find a wallet with $30$ Euro in it. The first child says: "If I keep the money for myself, I will have twice as much money as you both!" whereupon the second child proudly boasts: "And if I simply pocket the found money, I will have three times as much money as you both!" The third child can only smile smugly: "And if I take the money, I will be five times as rich as you two!" How much money did the children own before they found the wallet?
Let the Euro amounts which the three children owned before the find be denoted by $x$, $y$, and $z$, respectively. The statement of the first child can be translated into an algebraic equation as follows:

$x+30=2\left(y+z\right)⇔x-2y-2z=-30\mathrm{ }: \text{equation} \left(1\right) .$

Likewise, the statement of the second child can be translated into

$y+30=3\left(x+z\right)⇔-3x+y-3z=-30\mathrm{ }: \text{equation} \left(2\right) .$

And finally, the statement of the third child is translated into

$z+30=5\left(x+y\right)⇔-5x-5y+z=-30\mathrm{ }: \text{equation} \left(3\right) .$

So there arises a system of three linear equations in tree variables denoted here by $x$, $y$, and $z$.
The reader who is interested in the solution of this little puzzle will find it below worked out in detail using both the substitution method (see example 4.3.6) and the addition method (see example 4.3.8).
##### Info 4.3.2

A system of three linear equations in the three variables $x$, $y$, and $z$ has the following form:

$\begin{array}{cc}\multicolumn{1}{c}{{a}_{11}·x+{a}_{12}·y+{a}_{13}·z}& ={b}_{1} ,\hfill \\ \multicolumn{1}{c}{{a}_{21}·x+{a}_{22}·y+{a}_{23}·z}& ={b}_{2} ,\hfill \\ \multicolumn{1}{c}{{a}_{31}·x+{a}_{32}·y+{a}_{33}·z}& ={b}_{3} .\hfill \end{array}$

Here, ${a}_{11}$, ${a}_{12}$, ${a}_{13}$, ${a}_{21}$, ${a}_{22}$, ${a}_{23}$, ${a}_{31}$, ${a}_{32}$, and ${a}_{33}$ are the coefficients and ${b}_{1}$, ${b}_{2}$, and ${b}_{3}$ the right-hand sides of the system of linear equations.
Again, the system of linear equations is called homogeneous if the right-hand sides ${b}_{1}$, ${b}_{2}$, and ${b}_{3}$ are zero (${b}_{1}=0$, ${b}_{2}=0$, ${b}_{3}=0$). Otherwise, the system is called inhomogeneous.