Overview:
One of the ways to compare equations in linear systems is to see how many solutions both equations have in common. If both equations have no solutions in common, they are referred to as inconsistent. If only one ordered pair can solve both equations, the systems are called consistent, but if the equations have more than one point in common, they are dependent.
What Does It Mean to Have Solutions in Common?
A system of equations has solutions in common if there is at least one ordered pair that will solve both equations, even if there are many solutions that are not shared. For example, suppose one equation is x + y = 6 and another is x – y = 2. Do they have any solutions in common? There are many solutions to the equation x + y = 6. For example, if x equals 0, y equals 6, if x equals 1, y equals 5, if x equals 2, y equals 4 and so on. Similarly, in the second equation, if x equals 2, y equals 0, if x equals 3, y equals 1, and so on. Both equations do have one solution in common. If x equals 4 and y equals 2 in both equations, both equations have true solutions.
What Are Inconsistent Systems?
Inconsistent systems of linear equations have no solutions in common. When each equation is graphed on a coordinate plane, the lines are parallel. Suppose one equation is x – y = 8 and another equation is 5x – 5y = 25. The solutions for the first equation are ordered pairs such as {(16, 8), (15, 7), (14, 6), (13, 5), (12, 4) …}. The second equation can be simplified by dividing each member by 5, so that 5x/5 -5y/5 = 25/5. In other words, x – y = 5. They have no solutions in common.
What Are Consistent Systems?
Consistent systems have at least one solution in common. For example, the equations x + y = 6 and x – y = 2 have one solution in common, the ordered pair (4, 2) because 4 + 2 equals 6 and 4 – 2 equals 2. Similarly, the equations x + y = 12 and 3y = x have one solution in common, the ordered pair (9, 3), because 9 + 3 = 12 and 3(3) = 9.
What Are Dependent Systems?
Dependent systems have an infinite number of solutions in common. Both equations can be graphed on the same line. For example, suppose the equations were x – y = 5 and 5x – 5y = 25. The solution set of the first equation would include the points {(5, 0), (6, 1), (7-2), (8,3), …}. The solution set of the second equation would also include the points {(5, 0), (6, 1), (7, 2), (8,3),…}.
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