Question

Multiple Choice If a system of two linear equations in two variables is inconsistent, then the graphs of the lines in the system are ________. (a) intersecting (b) parallel (c) coincident (d) perpendicular

Short answer

The lines are parallel.

Step-by-step solution

Understand the Definition of 'Inconsistent'

A system of linear equations is considered inconsistent if there are no solutions that satisfy both equations simultaneously. This means the lines do not intersect at any point on the graph.

Analyze the Types of Line Relationships

Consider the relationships between two lines: they can be intersecting (one solution), parallel (no solution), coincident (infinitely many solutions), or perpendicular (one solution but at right angles).

Match Definition to Graphs

Since an inconsistent system has no solutions, the corresponding graph must feature lines that do not intersect at all. Among the given options, lines that do not intersect are defined as parallel.

Key concepts

linear equations

Linear equations form the core of many algebraic problems. A linear equation is an equation of the form \(ax + by = c\), where \(a\), \(b\), and \(c\) are constants. These equations graph as straight lines on a Cartesian plane.
When we talk about a 'linear system', we refer to a set of these equations working together. Usually, this means finding coordinates \((x, y)\) that satisfy both equations at the same time, essentially where the lines intersect.
Linear systems can have different types of solutions: a single unique solution (intersecting lines), no solution (parallel lines), or infinitely many solutions (coincident lines).

parallel lines

Parallel lines are an important concept when dealing with linear systems. Two lines are parallel if they never intersect, no matter how far they are extended. In terms of linear equations, this happens when the lines have the same slope but different y-intercepts.
For example, the lines given by \(y = 2x + 3\) and \(y = 2x - 4\) are parallel because their slopes (\(2\)) are identical, but their y-intercepts (\(3\) and \(-4\)) are different.
This characteristic of parallel lines leads to no points of intersection, which means, in a system of equations, it results in no solutions. Hence, such systems are termed 'inconsistent'.

no solutions

In the context of linear systems, 'no solutions' means there are no points \((x, y)\) that satisfy both linear equations simultaneously. This happens with parallel lines.
Mathematically, it is impossible to find a single pair \((x, y)\) that lies on both lines at the same time. This makes the system 'inconsistent', which is defined as a system with no common solutions.
Knowing how to identify an inconsistent system is crucial. Whenever you have two lines with the same slope but different y-intercepts, you can immediately tell that the system has no solutions.

graphical representation of lines

A graphical representation of lines helps visualize the relationship between them. By plotting linear equations on a Cartesian plane, you can see their intersections, parallels, or overlaps.
An intersection point means a unique solution, indicating that the system is consistent. If the lines overlap completely, there are infinitely many solutions, making the system also consistent but dependent.
When lines are parallel, you see no intersections, translating into 'no solutions' mathematically. Graphs effectively illustrate the concepts of intersecting, parallel, and coincident lines, helping to understand linear systems fully.