An Infinite Number of Solutions
If an inconsistent system has no solutions, another type of system, known as a dependent system, possesses an infinite number of solutions. The reason this occurs is that when you evaluate such systems, you find that you can express one variable in terms of the other. You have at hand such a system when you can multiply one of the equations in the system by some value that produces an equation that is the same as the other equation in the system. Equations that possess such a relationship with each other are known as dependent equations. To see how this can happen, consider this system of equations:
4x + 6y = 2
8x + 12y = 4
To make it easier to work with the two equations, reverse them:
8x + 12y = 4
4x + 6y = 2
Then to make it so that you can eliminate one of the variables, multiply the second equation by -2 to create an equivalent equation:
4x + 6y = 2 (multiply by -2)
The outcome of this activity is this equation:
-8x - 12y = -4
If you then add this equation to the first of the equations in the system, your activity proceeds along the following lines:
The system you are dealing with, then, consists of one equation expressed in two different ways, so to reach a solution for the system, you can solve 8x + 12y = 4 for x and y. One approach to this involves substitution and finding the solution for x. Accordingly, you might proceed in this way:
Given this finding, you can identify an ordered pair by using the value you possess for the y variable. This takes the following form:
Working from this basis, you can proceed to furnish any value you choose for y to arrive at the value of x. In this way, you can potentially generate an infinite number of ordered pairs. Among these are the following:
Exercise Set 11.5Explore these systems of equations and determine if they are dependent.
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