Introduction to linear equations
Posted by
Ravi Kumar at Wednesday, February 2, 2011
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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.