business mathematics

A linear equation in n unknowns x1, x2, · · · , xn is an equation of the form

a1x1 + a2x2 + · · · + anxn = b,

where a1, a2, . . . , an, b are given real numbers.

For example, with x and y instead of x1 and x2, the linear equation

2x + 3y = 6 describes the line passing through the points (3, 0) and (0, 2).

Similarly, with x, y and z instead of x1, x2 and x3, the linear equa-

tion 2x + 3y + 4z = 12 describes the plane passing through the points

(6, 0, 0), (0, 4, 0), (0, 0, 3).

A system of m linear equations in n unknowns x1, x2, · · · , xn is a family

of linear equations

a11x1 + a12x2 + · · · + a1nxn = b1

a21x1 + a22x2 + · · · + a2nxn = b2

..........

am1x1 + am2x2 + · · · + amnxn = bm.

We wish to determine if such a system has a solution, that is to find

out if there exist numbers x1, x2, · · · , xn which satisfy each of the equations

simultaneously. We say that the system is consistent if it has a solution.

Otherwise the system is called inconsistent.