Exercise Set 1.1 – No. 3

3. Using the notation of Formula:

\large {\color{Blue} \begin{matrix} a_{11}x_1 \: + \: a_{12}x_2 \: + \: ... \: + \: a_{1n}x_n \:= b_1 \\ a_{21}x_1 \: + \: a_{22}x_2 \: + \: ... \: + \: a_{2n}x_n \:= b_2 \\ \; \; \vdots \; \; \; \; \; \; \; \; \; \; \; \; \; \; \vdots \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \vdots \; \; \; \; \; \; \; \; \; \vdots \\ \: \: a_{m1}x_1 \: + \: a_{m2}x_2 \: + \: ... \: + \: a_{mn}x_n = b_m \\ \end{matrix} }

Write down a general linear system of

a) two equations in two unknowns.

b) three equations in three unknowns.

c) two equations in four unknowns.

a) A general linear system of two equations in two unknowns can be written
as:

\large {\color{blue}\begin{matrix} a_{11}x_1 \: + \: a_{12}x_2 = b_1 \\ a_{21}x_1 \: + \: a_{22}x_2 = b_2 \\ \end{matrix} }

where x1 and x2 are the unknowns.

b) A general linear system of three equations in three unknowns can be written
as:

\large {\color{blue}\begin{matrix} a_{11}x_1 \: + \: a_{12}x_2 + \: a_{13}x_3= b_1 \\ a_{21}x_1 \: + \: a_{22}x_2 + \: a_{23}x_3= b_2 \\ a_{31}x_1 \: + \: a_{32}x_2 + \: a_{33}x_3= b_3 \\ \end{matrix} }

where x1, x2 and x3 are the unknowns.

c) A general linear system of two equations in four unknowns can be written as:

\large {\color{blue}\begin{matrix} a_{11}x_1 \: + \: a_{12}x_2 + \: a_{13}x_3 + \: a_{14}x_4= b_1 \\ a_{21}x_1 \: + \: a_{22}x_2 + \: a_{23}x_3 + \: a_{24}x_4= b_2 \\ \end{matrix} }

where x1, x2, x3 and x4 are the unknowns.