Systems of linear equations - Essential Mathematics for Chemical Engineers

A system of linear equations has the form:

\begin{align} a_{11}x_{1} + \cdots + a_{1n}x_{n} &= b_{1} \\ \vdots \\ a_{m1}x_{1} + \cdots + a_{mn}x_{n} &= b_{n} \end{align}

(1)

There are n unknown variables, $(x_1, \cdots, x_n)$ and m equations. The m equations have coefficients $a_{mn}$ for each of the n variables and m right-hand side values $(b_1, \cdots, b_m)$ . These equations be rewritten using matrices and vectors:

\begin{bmatrix} a_{11} & \cdots & a_{1n} \\ \vdots & & \vdots \\ a_{m1} & \cdots & a_{mn} \end{bmatrix} \begin{bmatrix} \ x_{1} \\ \vdots \\ x_{n} \end{bmatrix} = \begin{bmatrix} b_{1} \\ \vdots \\ b_{m} \end{bmatrix}

(2)

or in shorthand:

\vv{A}\vv{x} = \vv{b}

(3)

A system of linear equations can have:

No solution (inconsistent equations)
One solution
Infinitely many solutions (undetermined)

For example, consider the following graphs of different systems of 2 equations:

$x+y = 1$ , $x+y = 0$
$One Solution$
This system of equations has no solution because the two lines are parallel and never intersect.
$x + y = 1$ , $2x - y = 0$
$One Solution$
This system of equations has one solution because the two lines intersect at a point.
$x+y = 1$ , $2x + 2y = 2$
$One Solution$
This system of equations has infinitely many solutions because the two lines are parallel and completely overlap.

3.2 Systems of linear equations