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Which System Has No Solution?

In mathematics, systems of equations play a crucial role in solving problems involving multiple variables. These systems consist of two or more equations that need to be solved simultaneously to find the values of the variables that satisfy all the equations. However, not all systems of equations have a solution. In this article, we will explore the concept of systems with no solution, understand the reasons behind it, and answer some frequently asked questions regarding this topic.

A system of equations is said to have no solution when the equations are inconsistent, meaning that there are no values for the variables that satisfy all the equations simultaneously. Mathematically, this can be represented as follows:

a₁x + b₁y = c₁

a₂x + b₂y = c₂

If the two lines represented by these equations are parallel, they will never intersect, indicating that there is no common solution for the system. Another possibility is that the lines are coincident, meaning they are the same line. In this case, there are infinite solutions since any point on the line will satisfy both equations. However, when the lines are distinct and do not intersect, the system has no solution.

There are several ways to determine if a system of equations has no solution. One of the most common methods is by examining the slopes of the lines. If the slopes are equal, but the y-intercepts are different, the lines are parallel and the system has no solution. If the slopes and y-intercepts are the same, the lines are coincident and the system has infinite solutions. Lastly, if the slopes are different, the lines intersect at a unique point, indicating that the system has one solution.

To illustrate this concept further, let’s consider an example. Suppose we have the following system of equations:

2x + 3y = 10

4x + 6y = 15

By rearranging the second equation, we can see that it is a multiple of the first equation:

2(2x + 3y) = 4x + 6y

4x + 6y = 10

Since the two equations are equivalent, they represent the same line. Therefore, any point that satisfies one equation will also satisfy the other. In this case, the system has infinite solutions.

Now, let’s address some frequently asked questions related to systems with no solution:

Q: Can a system of equations have more than one solution?

A: Yes, a system of equations can have infinite solutions or a unique solution, depending on the relationship between the equations.

Q: How can I determine if a system of three equations has no solution?

A: To determine if a system of three equations has no solution, you need to check if their corresponding planes intersect at a common point. If the planes are parallel or do not intersect, the system has no solution.

Q: Can a system of equations have no solution even if the equations are not parallel?

A: Yes, it is possible for a system of equations to have no solution even if the equations are not parallel. This occurs when the lines represented by the equations do not intersect.

Q: What is the significance of a system with no solution in real-life applications?

A: In real-life applications, a system with no solution may indicate an inconsistency or contradiction in the given data or conditions. It could mean that the problem statement is flawed or that the information provided is not sufficient to find a solution.

In conclusion, a system of equations has no solution when the lines represented by the equations do not intersect. This occurs when the lines are parallel or coincide with each other. Understanding and identifying systems with no solution is crucial in mathematics and real-life applications to avoid reaching incorrect or inconsistent conclusions.

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