The Fundamental Property of Consistent Systems in Linear Equations | Consistency for One Equals Consistency for All

if the system is consistent for some choice of constants, it is consistent for every choice of constants

In the context of linear equations, a consistent system refers to a system of equations that has at least one solution, meaning there is a set of values for the variables that satisfies all the equations

In the context of linear equations, a consistent system refers to a system of equations that has at least one solution, meaning there is a set of values for the variables that satisfies all the equations. On the other hand, an inconsistent system has no solutions, meaning there are no values for the variables that satisfy all the equations.

The statement “if the system is consistent for some choice of constants, it is consistent for every choice of constants” is a fundamental property of consistent systems. It states that if there exists a set of constant values that make the system consistent, then any other choice of constant values will also make the system consistent.

To understand why this is true, let’s consider a concrete example. Suppose we have a system of linear equations:

Equation 1: a*x + b*y = c
Equation 2: d*x + e*y = f

If we find a particular solution (values of x and y) that satisfies both equations, say (x_0, y_0), then we have:

a*x_0 + b*y_0 = c
d*x_0 + e*y_0 = f

Now, consider any other set of constants, let’s call them a’, b’, c’, d’, e’, and f’. We want to show that if the equations:

Equation 1′: a’*x + b’*y = c’
Equation 2′: d’*x + e’*y = f’

hold true, then they must have a solution as well. To do this, we can subtract Equation 1′ from Equation 1, and Equation 2′ from Equation 2:

(a*x_0 + b*y_0) – (a’*x + b’*y) = c – c’
(d*x_0 + e*y_0) – (d’*x + e’*y) = f – f’

Simplifying these equations, we obtain:

(a – a’)*x + (b – b’)*y = (c – c’)
(d – d’)*x + (e – e’)*y = (f – f’)

Now, notice that we have a new valid system of equations, and its constants are defined as the difference between the original constants (a – a’, b – b’, c – c’, d – d’, e – e’, f – f’). Since the original system is consistent, we know that there exists a solution (x_0, y_0) that satisfies the equations. Therefore, we can substitute these values into the new system of equations:

(a – a’)*x_0 + (b – b’)*y_0 = (c – c’)
(d – d’)*x_0 + (e – e’)*y_0 = (f – f’)

By doing this, we have found a set of variable values (x_0, y_0) that satisfies the new system. Thus, we conclude that if the system is consistent for some choice of constants, it is consistent for every choice of constants.

This property is useful because it allows us to determine the consistency of a system of equations by simply finding one solution. If a consistent solution can be found, then we know the system is consistent for all possible choices of constants.

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