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Which Linear Equation Has No Solution?

Linear equations are mathematical expressions that involve variables raised to the first power. They are characterized by their straight line graphs when plotted on a Cartesian plane. While most linear equations have solutions, there are certain cases where a linear equation has no solution. In this article, we will explore the conditions under which a linear equation has no solution and discuss some frequently asked questions on this topic.

Understanding Linear Equations:

Before delving into the concept of linear equations with no solution, let’s first understand what a linear equation is. A linear equation can be represented in the form “ax + by = c,” where “a,” “b,” and “c” are constants, and “x” and “y” are variables. The values of “x” and “y” that satisfy this equation represent the solution.

Cases with No Solution:

There are two main scenarios where a linear equation has no solution: parallel lines and overlapping lines.

1. Parallel Lines: When two linear equations have the same slope but different y-intercepts, they are considered parallel lines. These lines never intersect and, therefore, have no common solution. For instance, consider the equations y = 2x + 3 and y = 2x + 5. Both equations have the same slope (2), but different y-intercepts (3 and 5). When graphed, these lines will never intersect, indicating that there is no solution to this system of equations.

2. Overlapping Lines: Overlapping lines refer to two linear equations that are essentially the same line. In this case, any point on one line will also satisfy the other equation. When graphed, these lines coincide, resulting in an infinite number of solutions. For example, consider the equations 2x + 3y = 6 and 4x + 6y = 12. These equations represent the same line, as one equation can be obtained by multiplying the other by a constant. In this case, every point on this line will satisfy both equations, making the system have infinitely many solutions.

Frequently Asked Questions:

Q: Can a linear equation with one variable have no solution?

A: No, a linear equation with one variable will always have a solution. It can either have a unique solution or infinitely many solutions, depending on the equation.

Q: How can we determine if two lines are parallel without graphing them?

A: Two lines are parallel if their slopes are equal. If the slopes are different, the lines will intersect at a point.

Q: Is it possible for two linear equations to have more than one point of intersection?

A: No, two linear equations can intersect at most once. If they intersect at multiple points, they represent the same line.

Q: Are there any other scenarios where linear equations have no solution?

A: Apart from parallel lines and overlapping lines, linear equations can have no solution if the slopes are equal, but the y-intercepts are different. In this case, the lines are not parallel but will never intersect.

Q: Can a system of linear equations have both no solution and infinitely many solutions?

A: No, a system of linear equations can only have one of these possibilities: a unique solution, no solution, or infinitely many solutions. It cannot have both no solution and infinitely many solutions simultaneously.

In conclusion, certain conditions can lead to linear equations having no solution. Parallel lines, where two equations have the same slope but different y-intercepts, and overlapping lines, where two equations are essentially the same line, are the main cases with no solution. Understanding these scenarios can help in solving linear equations and analyzing their solutions.

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