Identifying Systems Of Equations With No Solution

In the realm of mathematics, specifically within the study of linear equations, a crucial concept is the solution to a system of equations. A system of linear equations represents a set of two or more equations sharing the same variables. The solution to such a system is the set of values for the variables that simultaneously satisfy all equations in the system. However, not all systems of linear equations possess a solution. This article delves into the question of which system has no solution, exploring the conditions that lead to inconsistent systems and providing a comprehensive analysis of the given examples.

Understanding Solutions to Systems of Linear Equations

Before we pinpoint the system with no solution, it's essential to grasp the different scenarios that can arise when solving systems of linear equations. Graphically, each linear equation represents a straight line. When dealing with a system of two equations in two variables (typically x and y), there are three possible outcomes:

1. Unique Solution: The lines intersect at a single point. This point represents the unique solution to the system, where the x and y coordinates satisfy both equations.
2. Infinitely Many Solutions: The lines coincide, meaning they are essentially the same line. In this case, every point on the line satisfies both equations, resulting in infinitely many solutions.
3. No Solution: The lines are parallel and never intersect. This signifies that there is no point (x, y) that can satisfy both equations simultaneously, indicating an inconsistent system.

Inconsistent systems are the focus of our discussion, as they represent the scenario where no solution exists. Parallel lines are the key indicator of an inconsistent system. Mathematically, parallel lines have the same slope but different y-intercepts. This means that the coefficients of x and y in the equations are proportional, but the constant terms are not.

Analyzing the Given Systems of Equations

Now, let's examine the provided systems of equations to determine which one has no solution. We'll analyze each system by comparing the slopes and y-intercepts of the lines represented by the equations.

System 1:

y = -3x + 8
6x + 2y = -4.5

To analyze this system, we need to express both equations in the slope-intercept form (y = mx + b), where m is the slope and b is the y-intercept. The first equation is already in this form. Let's rewrite the second equation:

6x + 2y = -4.5
2y = -6x - 4.5
y = -3x - 2.25

Now we have both equations in slope-intercept form:

y = -3x + 8
y = -3x - 2.25

Notice that both lines have the same slope (m = -3) but different y-intercepts (8 and -2.25). This indicates that the lines are parallel and will never intersect. Therefore, this system has no solution.

System 2:

y = 9x + 6.25
-18x + 2y = 12.5

Let's convert the second equation to slope-intercept form:

-18x + 2y = 12.5
2y = 18x + 12.5
y = 9x + 6.25

Now we have:

y = 9x + 6.25
y = 9x + 6.25

Both equations are identical, representing the same line. This means the lines coincide, and there are infinitely many solutions.

System 3:

y = 4.5x - 5
-3x + 2y = 6

Convert the second equation to slope-intercept form:

-3x + 2y = 6
2y = 3x + 6
y = 1.5x + 3

Now we have:

y = 4.5x - 5
y = 1.5x + 3

The slopes (4.5 and 1.5) are different. This indicates that the lines intersect at a single point, meaning the system has a unique solution.

System 4:

y = 3x + 9
x + 8y = 12.3

Convert the second equation to slope-intercept form:

x + 8y = 12.3
8y = -x + 12.3
y = -0.125x + 1.5375

Now we have:

y = 3x + 9
y = -0.125x + 1.5375

The slopes (3 and -0.125) are different. This signifies that the lines intersect at a single point, indicating a unique solution.

Conclusion: The Inconsistent System Revealed

Through our analysis, we've identified that the first system of equations:

y = -3x + 8
6x + 2y = -4.5

has no solution. This is because the two equations represent parallel lines, characterized by the same slope but different y-intercepts. This understanding of inconsistent systems is crucial in solving linear equations and interpreting their graphical representations. When encountering systems of equations, always consider the possibility of parallel lines, as they signify the absence of a solution.

In summary, the ability to identify systems with no solution is a fundamental skill in mathematics. By understanding the relationship between slopes and y-intercepts, we can effectively determine whether a system of linear equations will have a unique solution, infinitely many solutions, or no solution at all. This knowledge is not only valuable in academic settings but also in various real-world applications where linear equations are used to model and solve problems.