Systems Of Equations With No Solution An Explanation

When exploring systems of linear equations, a critical concept to grasp is the scenario where no solution exists. This occurs when the lines represented by the equations are parallel and never intersect. To delve into this, let's consider the given equation and analyze the conditions under which a second equation would result in a system with no solution. This article aims to provide a comprehensive understanding of this concept, guiding you through the process of identifying such equations and the mathematical principles behind them.

The Given Equation: y = 8x + 7

We are given the equation y = 8x + 7. This is a linear equation in slope-intercept form, where the slope (m) is 8 and the y-intercept (b) is 7. The slope determines the steepness and direction of the line, while the y-intercept is the point where the line crosses the y-axis. To have no solution in a system of two linear equations, the second line must be parallel to this line but have a different y-intercept. Parallel lines, by definition, have the same slope but different y-intercepts. This ensures they never intersect, indicating that there are no common points (solutions) that satisfy both equations simultaneously. Therefore, when we're trying to find a second equation that results in no solution, we need to focus on equations that have a slope of 8 but a y-intercept that is different from 7. This is the fundamental principle that governs the existence of solutions in a system of linear equations. Understanding this principle is crucial for solving problems involving systems of equations and determining their solutions.

Identifying Equations with No Solution

To find another equation that results in no solution when paired with y = 8x + 7, we need an equation with the same slope but a different y-intercept. This is because parallel lines have the same slope but intersect the y-axis at different points. Let's analyze the provided options, focusing on rewriting them in slope-intercept form (y = mx + b) to easily identify their slopes and y-intercepts. The slope is the coefficient of x (m), and the y-intercept is the constant term (b). Remember, for there to be no solution, the slopes must be equal, and the y-intercepts must be different. This ensures the lines are parallel but not overlapping. We will go through each option, transforming them if necessary, and comparing their slopes and y-intercepts with the given equation. This methodical approach will allow us to pinpoint the equation that, when paired with y = 8x + 7, creates a system with no solution. This exercise is a practical application of the theoretical understanding of systems of equations and their geometric representation as lines in a coordinate plane.

Analyzing Option A: 2y = 16x + 14

Option A presents the equation 2y = 16x + 14. To determine if this equation results in no solution when paired with y = 8x + 7, we must first convert it into slope-intercept form (y = mx + b). This involves isolating y on one side of the equation. We achieve this by dividing both sides of the equation by 2:

(2y) / 2 = (16x + 14) / 2

This simplifies to:

y = 8x + 7

Upon converting the equation, we observe that it is identical to the original equation, y = 8x + 7. This means that the two equations represent the same line. When two equations represent the same line, they have infinite solutions because every point on the line satisfies both equations. Therefore, option A does not result in a system with no solution. Instead, it represents a system with infinitely many solutions. This highlights the importance of not just looking at the surface appearance of an equation but also transforming it into a standard form for accurate comparison. Understanding this distinction is crucial for correctly identifying the nature of solutions in a system of equations.

Analyzing Option B: y = 8x - 7

Option B gives us the equation y = 8x - 7. This equation is already in slope-intercept form (y = mx + b), making it easy to identify the slope and y-intercept. Here, the slope (m) is 8, and the y-intercept (b) is -7. Comparing this with the original equation, y = 8x + 7, we notice that the slopes are the same (both are 8), but the y-intercepts are different (7 and -7). This is precisely the condition for two lines to be parallel and thus have no intersection points. Since the lines are parallel and do not intersect, the system of equations formed by y = 8x + 7 and y = 8x - 7 has no solution. Therefore, option B is a potential candidate for the correct answer. This analysis underscores the significance of the slope-intercept form in quickly determining the relationship between two lines and the nature of their solutions. The ability to readily identify the slope and y-intercept is a valuable skill in solving problems involving systems of linear equations.

Analyzing Option C: y = -8x + 7

Option C presents the equation y = -8x + 7. This equation is already in slope-intercept form (y = mx + b), which allows for a straightforward identification of the slope and y-intercept. In this case, the slope (m) is -8, and the y-intercept (b) is 7. When comparing this equation with the given equation, y = 8x + 7, we observe that the slopes are different (-8 versus 8), and the y-intercepts are the same (both are 7). Since the slopes are different, the lines are not parallel; they will intersect at some point. Therefore, the system of equations formed by y = 8x + 7 and y = -8x + 7 will have a single unique solution. This means that option C does not result in a system with no solution. The fact that the y-intercepts are the same indicates that the point of intersection will lie on the y-axis, specifically at the point (0, 7). This analysis reinforces the importance of carefully examining both the slope and y-intercept when determining the nature of solutions in a system of linear equations.

Analyzing Option D: 2y = -16x - 14

Option D provides the equation 2y = -16x - 14. To properly analyze this equation and compare it with the given equation, y = 8x + 7, we need to convert it into slope-intercept form (y = mx + b). This involves isolating y on one side of the equation. We can achieve this by dividing both sides of the equation by 2:

(2y) / 2 = (-16x - 14) / 2

Simplifying the equation, we get:

y = -8x - 7

Now that the equation is in slope-intercept form, we can easily identify the slope and y-intercept. The slope (m) is -8, and the y-intercept (b) is -7. Comparing this with the original equation, y = 8x + 7, we see that the slopes are different (-8 versus 8), and the y-intercepts are also different (7 versus -7). Since the slopes are different, the lines represented by these equations are not parallel. They will intersect at a single point, meaning the system of equations will have one unique solution. Thus, option D does not result in a system with no solution. This analysis highlights the crucial role of converting equations to slope-intercept form to accurately determine their properties and relationships.

Conclusion: The Equation Resulting in No Solution

After analyzing all the options, we can confidently conclude that the equation that results in no solution when paired with y = 8x + 7 is option B: y = 8x - 7. This is because option B has the same slope (8) but a different y-intercept (-7) compared to the original equation. Equations with the same slope but different y-intercepts represent parallel lines, which never intersect, resulting in a system with no solution. Understanding the relationship between the slopes and y-intercepts of linear equations is fundamental to determining the nature of solutions in a system of equations. This exercise not only helps in identifying equations with no solution but also reinforces the core principles of linear algebra and the geometric interpretation of linear equations.

Final Answer

The final answer is B. y = 8x - 7.