System Of Equations Analysis Determine True Statements

In the realm of mathematics, particularly in linear algebra, systems of equations form a foundational concept. These systems, comprising two or more equations, describe relationships between variables and often represent real-world scenarios that can be modeled mathematically. Solving these systems involves finding values for the variables that satisfy all equations simultaneously. This exploration delves into the intricacies of analyzing a specific system of two linear equations, aiming to determine key properties such as the number of solutions and the geometric relationship between the lines they represent. Understanding these properties is crucial for various applications, including modeling supply and demand in economics, determining optimal resource allocation in operations research, and even simulating physical phenomena in engineering.

The given system of equations is:

$ y = \frac{1}{3}x - 4 $$ 3y - x = -7 $

To analyze this system effectively, we need to consider several aspects, including the slopes and y-intercepts of the lines represented by these equations. The slope of a line provides information about its steepness and direction, while the y-intercept indicates the point where the line crosses the vertical axis. By comparing these characteristics, we can ascertain whether the lines intersect, are parallel, or coincide, which in turn determines the number of solutions the system possesses. Furthermore, transforming the equations into slope-intercept form ($y = mx + b$), where m represents the slope and b represents the y-intercept, simplifies the analysis process. In this article, we will meticulously examine each equation, transform them as needed, and then draw conclusions about the system's nature.

Transforming Equations to Slope-Intercept Form

The first step in analyzing the given system is to express both equations in slope-intercept form, which is $y = mx + b$. This form makes it easy to identify the slope (m) and y-intercept (b) of each line. The first equation, $y = \frac{1}{3}x - 4$, is already in slope-intercept form. Here, the slope is $\frac{1}{3}$ and the y-intercept is -4. This means that the line rises one unit vertically for every three units it moves horizontally, and it crosses the y-axis at the point (0, -4).

The second equation, $3y - x = -7$, needs to be rearranged. To isolate y, we first add x to both sides of the equation, resulting in $3y = x - 7$. Next, we divide both sides by 3 to obtain $y = \frac{1}{3}x - \frac{7}{3}$. Now, this equation is also in slope-intercept form. We can see that the slope is $\frac{1}{3}$ and the y-intercept is -$\frac{7}{3}$ (or approximately -2.33). This line has the same steepness as the first line (since they have the same slope) but crosses the y-axis at a different point.

Having both equations in slope-intercept form allows for a direct comparison of their slopes and y-intercepts. This comparison is crucial for determining the geometric relationship between the lines and, consequently, the number of solutions the system has. In the next section, we will delve into this comparison and explore the implications for the system's solutions.

Comparing Slopes and Y-Intercepts

Now that both equations are in slope-intercept form, we can compare their slopes and y-intercepts to understand the relationship between the lines they represent. The first equation, $y = \frac{1}{3}x - 4$, has a slope of $\frac{1}{3}$ and a y-intercept of -4. The second equation, $y = \frac{1}{3}x - \frac{7}{3}$, also has a slope of $\frac{1}{3}$, but its y-intercept is -$\frac{7}{3}$.

The fact that the slopes are the same ($\frac{1}{3}$) indicates that the lines are parallel. Parallel lines, by definition, have the same slope and never intersect. This geometric property has significant implications for the system of equations. Since the lines never intersect, there is no point (x, y) that satisfies both equations simultaneously. In other words, there is no common solution to the system.

However, it's important to note that parallel lines can either be distinct (as in this case) or coincident. Coincident lines are essentially the same line, meaning they have the same slope and the same y-intercept. In this scenario, the system would have infinitely many solutions because every point on the line would satisfy both equations. But in our case, the y-intercepts are different (-4 and -$\frac{7}{3}$), so the lines are distinct and parallel. This difference in y-intercepts confirms that the lines are not overlapping but are running alongside each other without ever meeting.

The implications of parallel lines extend beyond just the number of solutions. It also provides valuable insight into the nature of the problem being modeled by the system of equations. For instance, if these equations represented supply and demand curves in economics, the parallelism would indicate a situation where there is no equilibrium point, meaning the market forces are not balanced.

Determining the Number of Solutions

Based on our analysis of the slopes and y-intercepts, we can now definitively determine the number of solutions for the system of equations. As we established, both lines have the same slope ($\frac{1}{3}$) but different y-intercepts (-4 and -$\frac{7}{3}$). This means the lines are parallel and distinct; they will never intersect.

In the context of systems of linear equations, the number of solutions corresponds to the number of points where the lines intersect. Since parallel lines do not intersect, the system has no solution. This is a crucial concept in linear algebra and has practical implications in various fields. For example, in linear programming, if a system representing constraints has no solution, it indicates that the problem is infeasible and there is no combination of variables that satisfies all the constraints.

Contrast this with systems where the lines intersect at a single point, which have one unique solution, or systems where the lines are coincident, which have infinitely many solutions. Understanding these different scenarios is fundamental to solving and interpreting systems of equations. The ability to quickly identify the number of solutions by analyzing slopes and y-intercepts is a valuable skill in mathematical problem-solving.

In summary, the system of equations:

$ y = \frac{1}{3}x - 4 $$ 3y - x = -7 $

has no solution because the lines are parallel and distinct. This conclusion is reached by transforming the equations into slope-intercept form, comparing their slopes and y-intercepts, and recognizing the geometric implications of parallel lines.

Identifying True Statements about the System

Now that we have thoroughly analyzed the system of equations, we can evaluate the given statements to determine which ones are true. The system we are considering is:

$ y = \frac{1}{3}x - 4 $$ 3y - x = -7 $

Let's examine each statement:

A. The system has one solution.

We have already established that the lines are parallel and do not intersect. Therefore, this statement is false. A system has one solution only when the lines intersect at a single point.

B. The system consists of parallel lines.

As determined earlier, both lines have the same slope ($\frac{1}{3}$) but different y-intercepts. This confirms that the lines are indeed parallel. Thus, this statement is true. Parallel lines are a key characteristic of this system, leading to its lack of a solution.

C. Both lines have the same slope.

By converting both equations to slope-intercept form, we found that both lines have a slope of $\frac{1}{3}$. This confirms that this statement is true. The equal slopes are the reason why the lines are parallel.

Based on our analysis, the two true statements about the system are:

• B. The system consists of parallel lines.
• C. Both lines have the same slope.

These statements accurately describe the geometric relationship between the lines represented by the equations and are the correct choices for this problem.

Conclusion

In conclusion, the analysis of the system of linear equations:

$ y = \frac{1}{3}x - 4 $$ 3y - x = -7 $

reveals important insights about its properties. By converting the equations to slope-intercept form, we identified that both lines have the same slope ($\frac{1}{3}$) but different y-intercepts. This critical observation led us to the conclusion that the lines are parallel and distinct. As a result, the system has no solution because the lines never intersect.

This exercise underscores the significance of understanding the relationship between the slopes and y-intercepts of linear equations and how they determine the nature of the system. Parallel lines, intersecting lines, and coincident lines each have distinct characteristics that dictate the number of solutions a system possesses. Being able to quickly analyze these properties is a fundamental skill in mathematics and has wide-ranging applications in various fields.

Furthermore, this analysis highlights the importance of careful and methodical problem-solving. By breaking down the problem into smaller steps – converting to slope-intercept form, comparing slopes and y-intercepts, and interpreting the geometric implications – we were able to confidently determine the correct answers. This approach is applicable to a wide range of mathematical problems and is a valuable tool for students and professionals alike.

Therefore, the true statements about the system are that it consists of parallel lines and both lines have the same slope. This comprehensive analysis provides a clear understanding of the system's characteristics and reinforces the key concepts of linear equations and their solutions.