Equivalent Systems of Equations

Introduction:

An equivalent system of equation is a system of equations in which both systems have same solution but they may have different numbers of equations.

Systems of Linear Equations

It is a set of algebraic expressions in the form:


a11x1 + a12x2 + .....................+a1nxn = b1


a21x1 + a22x2 + .....................+a2nxn = b2


am1x1 + am2x2 + .....................+amnxn = bm


xi are the unknowns, (i = 1, 2, ..., n).
aij are the coefficients, (i = 1, 2, ..., m), (j = 1, 2, ..., n).
bi are the independent terms, (i = 1, 2, ..., m).
m, n ; m > n, or, m = n, or, m < n.
The number of equations need not equal the number of unknowns.
aij and bi  .
When n is less, it is usual to assign the unknowns with the letters x, y, z, t, ...
When bi = 0, for all i, the system is called homogeneous.

Equivalent systems of equations are obtained by elimination if:


The value of coefficients is zero.
Two equal rows are present.
Two rows are proportional to each other.
A row is formed by linear combination of others. Looking out for more help on Algebra Mixture Problems in algebra by visiting listed websites.

Equivalence Criteria

The resulting system is equivalent if both members of an equation of a system are added or subtracted by the same expression.
The resultant system is equivalent, if both members of the equations of a system are multiplied or divided by a number other than zero,
The resultant system is equivalent, if an equation of a system is added or reduced by another equation of the same system.
The resultant system is equivalent, if an equation in a system is replaced by another equation that results from adding the equations of a system previously multiplied or divided by nonzero numbers,
If the order of the unknowns of a system or order of the equations is changed, it is another equivalent system.