The inconsistent system of linear equations is one for which there is no solution. When represented on a graph, the two lines are parallel and do not overlap.
When solving a system of equations, the solution is either one point, no solution or infinitely many solutions. When two lines intersect, they intersect at a single point and the equations are independent. When the answer to an equation is always true, such as 3=3, the equations are dependant and a single line represents both equations on the graph. However, when solving the system gives inaccuracy or untruth, such as 0=5, the system is inconsistent and has no solution.
By definition, parallel lines are those that never intersect. Parallel lines have the same slope. Mathematicians use the point slope form of an equation, y=mx+b, to determine the slope of a line and its x-intercept. In this equation, 'm' represents the intercept. When the equations in an inconstant set are arranged in this form, the slope is the same, while the intercept, b, differs.
The slope of a line is also calculated using any two points on the line and the equation m=(y1-y2)/(x1-x2) for all lines except vertical ones. In vertical lines, all the x-values are the same, resulting in division by zero when using the equation. While vertical lines have no slope, they are still defined as parallel.