What makes an equation consistent or inconsistent

These are referred to as Consistent Systems of Equations, meaning that for a given system, there exists one solution set for the different variables in the system or infinitely many sets of solution. In other words, as long as we can find a solution for the system of equations, we refer to that system as being consistent.

For a two variable system of equations to be consistent the lines formed by the equations have to meet at some point or they have to be parallel. For a three variable system of equations to be consistent, the equations formed by the equations must meet two conditions:.

Given that such systems exist, it is safe to conclude that Inconsistent systems should exist as well, and they do. Inconsistent Systems of Equations are referred to as such because for a given set of variables, there in no set of solutions for the system of equations. The equations in a two variable system of equations are linear and hence can be thought of as equations of two lines.

When these two lines are parallel, then the system has infinitely many solutions. That means there is only one solution to the system. The point of intersection appears to be 1, 2.

Read the point from the graph as accurately as possible. Check the values in both equations. Substitute 1 for x and 2 for y. Since 1, 2 is a solution for each of the equations in the system, it is the solution for the system. The two equations graph as the same line. So every point on that line is a solution for the system of equations.

The system is graphed correctly below. This means it cannot be a solution for the system. Graphing a Real-World Context. Graphing a system of equations for a real-world context can be valuable in visualizing the problem. The number of 2-point shots she made was one greater than the number of 3-point shots she made.

How many of each type of basket did she score? Assign variables to the two unknowns — the number of each type of shots. Calculate how many points are made from each of the two types of shots.

Write an equation using information given in the problem. Write a second equation using additional information given in the problem. Now you have a system of two equations with two variables. Graph both equations on the same axes. The two lines intersect, so they have only one point in common. The point of intersection appears to be 4, 3. Read the point of intersection from the graph. Check 4, 3 in each equation to see if it is a solution to the system of equations.

Cheryl made 4 two-point baskets and 3 three-point baskets. Andres was trying to decide which of two mobile phone plans he should buy. To examine the difference in plans, he made a graph:. If he plans to talk on the phone for about 70 minutes per month, which plan should he purchase? Look at the graph. The number of minutes is listed on the x -axis. Andres should buy theTalkALot plan.

Since TalkALot costs less at 70 minutes, Andres should buy that plan. Note that if the estimate had been incorrect, a new estimate could have been made. They have no solutions in common. Consistent systems have at least one solution in common.

Dependent systems have an infinite number of solutions in common. Both equations can be graphed on the same line. Interested in algebra tutoring services? Learn more about how we are assisting thousands of students each academic year. SchoolTutoring Academy is the premier educational services company for K and college students. We offer tutoring programs for students in K, AP classes, and college.

If a consistent system has exactly one solution, it is independent. If a consistent system has an infinite number of solutions, it is dependent. When you graph the equations, both equations represent the same line. If a system has no solution, it is said to be inconsistent.