Steps in solving problems involving systems of linear equations?

Answer 1

The answer depends on the level of your knowledge. The High level, simple answer is first. The Low level slog follows:

HIGH LEVEL, SIMPLE

Suppose you have n equations of the form

a11x1 + a12x2 + ... + a1nxn = bn where

the as are coefficients,

x1, x2, ... xn are the unknown variables

and

b1, b2, ... bn are the constants.

Write the n linear equations in n unknowns in the form Ax= b

where

A is an n*n matrix of coefficients

x is the n*1 matrix of the unknown variables

and

b is the n*1 matrix of the constants.

Find the inverse of A.

Then x = A-1b.

The above method works if the system has a unique solution. If the n equations are not independent, you will need to use a generalised inverse and that starts to get rather complicated. If they are inconsistent, then neither the inverse nor generalised inverse will be found.

LOW LEVEL SLOG

Use the first equation to express x1 in terms of the other variables. Substitute this value for x1 in the remaining n-1 equations. You now have n-1 equations in n-1 unknown variables.

Use the first of the new equations to express x2 in terms of the other variables. Substitute in remaining equations. You now have n-2 equations in n-2 unknown variables.

Continue until you have 1 equation in 1 unknown.

That will be of the form pxn = q so that xn = q/p.

Substitute this value into one of the equations at the 2-equations-in-2-unknowns stage. That will give you xn-1.

Work your way back to the top.

The two methods are equivalent. There are shortcuts available for matrix inversion (eg using determinants), but these are too complicated to go into here.