Question

Describe the graph of a linear system that has the given number of solutions. Sketch an example. no solution

Short answer

A linear system with no solution would be graphically represented by two parallel lines. An example can be the lines of the linear equations \(y = 2x + 3\) and \(y = 2x + 1\).

Step-by-step solution

Understanding Linear System With No Solution

A linear system with no solution is a system where the graphs of the equations do not intersect at any point. Meaning the lines represented by the equations are parallel to each other. So given a condition of a linear system with no solution, the graphs would be two parallel lines.

Sketching the example

On a two-dimensional graph, sketch two lines with the same slope but different y-intercepts. For instance, let's consider two lines represented by equations \(y = 2x + 3\) and \(y = 2x + 1\). Both lines have the same slope of 2 but with different y-intercepts: line 1 intersect the y-axis at point (0,3) and line 2 intersect the y-axis at point (0,1). Thus, these two lines will never intersect each other in the Euclidean plane, indicating that the system of equations has no solution.

Key concepts

Understanding Parallel Lines

Parallel lines are a fundamental concept in geometry and an integral part of understanding why some linear systems have no solution. Imagine walking on two railroad tracks that never meet, no matter how far you walk. That's the essence of parallel lines.

By definition, parallel lines are lines in a plane that are always the same distance apart and never intersect. In the context of a linear system, if we have two linear equations representing two lines, and these lines are parallel, it means that they have the same slope but different y-intercepts. The slope essentially describes the steepness or incline of the line, while the y-intercept tells us where the line crosses the y-axis.

Consider the following visual guideline for identifying parallel lines on a graph:

• Both lines will appear to move in the same direction.
• If extended indefinitely, the lines will never touch or cross over each other.
• The angles connecting a transversal line with our parallel lines will be congruent, indicating the lines run perfectly in sync, never converging or diverging.

For a linear system that has no solution, the representation on a graph would be that of two parallel lines. This signifies that there is no single point that satisfies both equations simultaneously, thus leaving us with a system that cannot be solved.

Deciphering the Slope of a Line

The concept of the slope is vital in understanding linear relationships and is key in distinguishing between different types of linear systems. The slope of a line measures its steepness, it's a numerical value that represents the rate at which the line ascends or descends as you move from left to right across the graph.

A simple way to calculate the slope of a line is by using the formula: \[ m = \frac{(y2 - y1)}{(x2 - x1)} \] where \((x1, y1)\) and \((x2, y2)\) are any two different points lying on the line. If the slope value, denoted as 'm', is positive, the line inclines upwards; if 'm' is negative, the line declines. If two lines have the same slope, they are either the exact same line or they are parallel lines.

For instance, if we were to analyze the lines given by the equations \(y = 2x + 3\) and \(y = 2x + 1\), both would have a slope of '2'. This consistent slope tells us that the lines have the same angle of inclination, and because their y-intercepts are not the same, we deduce that the lines are distinct and parallel to each other.

Grasping the Y-intercept

The y-intercept of a line is a simple concept, yet it is crucial for graphing linear equations and for understanding the make-up of a linear system. The y-intercept occurs where the graph of the line crosses the y-axis. Geometrically, it's the point where the x-coordinate is zero.

Every linear equation in the slope-intercept form \(y = mx + b\) has a y-intercept represented by 'b'. This value is a constant that determines the exact location where the line will touch the y-axis. It's like pinning a tail on the y-axis; that pin represents the y-intercept.

When analyzing two lines with the same slope but different y-intercepts, we can immediately tell that the lines will never intersect, leading to a system of equations with no solutions. For example, take the lines \(y = 2x + 3\) and \(y = 2x + 1\). Their y-intercepts are 3 and 1, respectively, which means they touch the y-axis at different points, creating parallel lines that will never meet, hence, no shared solutions.