Problem 114 - Algebra and Trigonometry

Question

What is an inconsistent equation? Give an example.

Step-by-Step Solution

Answer

An inconsistent equation is a type of system of equations that has no solution. This happens when the represented lines on a graph do not intersect. An example of an inconsistent equation is: \[2x + 3y = 10\] and \[4x + 6y = 12\]. Despite having the same slopes, they will never intersect because they don't have the same y-intercepts.

Step 1: Definition of inconsistent equations

In algebra, an inconsistent equation is one that forms a system of equation which has no solution. This happens when the lines, represented on a graph by these equations, are parallel and do not intersect each other. A system of linear equations can be inconsistent if the slopes of the lines they represent are equal.

Step 2: Providing an Example

An example of inconsistent equations is: \[ 2x+3y=10 \] \[ 4x+6y=12 \] If the equations are graphed, they would produce parallel lines.

Step 3: Explaining the Example

In an inconsistent system, regardless of the number of equations or variables in the system, it's impossible to find a solution that satisfies all the equations at once. In the given example, both lines have the same slope (i.e., their ratio of y over x are equal: 3/2 = 6/4). But their y-intercepts are not equal (10/3 ≠ 12/6). So, the lines are parallel and will never intersect. Thus, there's no point (x, y) that satisfies both equations at the same time, making them inconsistent.