What makes a system consistent? Inconsistent?
Consistent if there is at least 1 solution (there can be infinite) Inconsistent if there is no solution
A system of equations is consistent if it has at least one solution and is inconsistent if it has no solution.
A system of equations is consistent when all the equations in the system have at least one point in common. This means that all the equations intersect at a single point, forming a consistent solution. If there are two or more equations in the system, they must be linearly independent so that they do not contradict each other and have a common solution. In other words, the equations should not be parallel or identical as they will not have a common solution.
On the other hand, a system of equations is inconsistent when there is no solution that satisfies all the equations in the system. Geometrically, this means that the equations represent lines that do not intersect at any point, forming parallel lines. For instance, consider the system of equations x + y = 3 and 2x + 2y = 6. These equations represent two parallel lines, and therefore, there is no common solution. The absence of a common solution makes the system of equations inconsistent.
In summary, a system of equations is consistent when all equations have at least one solution in common and is inconsistent when there is no solution that satisfies all the equations in the system.
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