What is an inconsistent equation? Give an example.

Verified step by step guidance

1. An inconsistent equation is a type of equation or system of equations that has no solution. This occurs when the equations in the system contradict each other, meaning there is no set of values that satisfies all the equations simultaneously.
2. For example, consider the system of two linear equations: y = 2x + 3 and y = 2x - 5. These two equations represent two parallel lines because they have the same slope (2) but different y-intercepts (3 and -5).
3. Since parallel lines never intersect, there is no point (x, y) that satisfies both equations. This makes the system inconsistent.
4. To verify inconsistency algebraically, set the two equations equal to each other: 2x + 3 = 2x - 5. Subtract 2x from both sides to get 3 = -5, which is a contradiction.
5. The contradiction (3 = -5) confirms that the system is inconsistent, meaning there is no solution.

Key Concepts

Inconsistent Equations

An inconsistent equation is a mathematical statement that has no solution. This occurs when the equations represent parallel lines in a graph, meaning they never intersect. For example, the equations '2x + 3 = 7' and '2x + 3 = 5' are inconsistent because they imply different values for the same variable.

Graphical Representation

Graphical representation of equations helps visualize their relationships. Inconsistent equations can be represented as lines on a coordinate plane that do not meet. Understanding how to graph these equations allows students to see why certain equations are inconsistent, reinforcing the concept through visual means.

Systems of Equations

A system of equations consists of two or more equations that share variables. An inconsistent system occurs when there are no common solutions among the equations. Recognizing the nature of systems, whether consistent or inconsistent, is crucial for solving problems in algebra and understanding the relationships between different equations.