System Of Equations Analysis Determining True Statements For Y=(1/3)x-4 And 3y-x=-7

Jul 13, 2025 by ADMIN 84 views

$Understanding Systems of Equations A Detailed Analysis of ${ y=\frac{1}{3} x-4 }$ and ${ 3y-x=-7 }$$

In the realm of mathematics, systems of equations represent a fundamental concept with far-reaching applications. These systems, composed of two or more equations, provide a framework for modeling and solving real-world problems across various disciplines. In this comprehensive analysis, we will delve into the intricacies of a specific system of equations, carefully examining its properties and solutions. Our focus will be on the system:

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

$ 3y - x = -7 $

We will explore the nature of the lines represented by these equations, determine the number of solutions the system possesses, and discuss the significance of these solutions in the broader context of mathematical problem-solving. Understanding systems of equations is crucial not only for academic pursuits but also for practical applications in fields such as engineering, economics, and computer science.

Analyzing the Given System of Equations

To gain a thorough understanding of the system, we will systematically analyze each equation. The first equation,

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

is presented in slope-intercept form, which provides immediate insights into the line's characteristics. The slope, denoted by the coefficient of x, is ${ \frac{1}{3} }$ , and the y-intercept, where the line crosses the y-axis, is -4. This information allows us to visualize the line's orientation and position on the coordinate plane.

The second equation,

$ 3y - x = -7 $

requires a bit more manipulation to reveal its properties. To bring it into slope-intercept form, we need to isolate y. We begin by adding x to both sides of the equation:

$ 3y = x - 7 $

Next, we divide both sides by 3:

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

Now, the equation is in slope-intercept form, making it clear that the slope is ${ \frac{1}{3} }$ and the y-intercept is ${ -\frac{7}{3} }$ . Comparing the two equations, we observe that they share the same slope but have different y-intercepts. This observation is crucial in determining the nature of the lines and the system's solutions.

Determining the Number of Solutions

The number of solutions a system of equations possesses is directly related to the geometric relationship between the lines represented by the equations. There are three possible scenarios:

One Solution: The lines intersect at a single point, indicating a unique solution that satisfies both equations.
No Solution: The lines are parallel and never intersect, implying that there is no solution that satisfies both equations simultaneously.
Infinitely Many Solutions: The lines are coincident, meaning they overlap completely. In this case, every point on the line represents a solution to the system.

In our system, the lines have the same slope ( ${ \frac{1}{3} }$ ) but different y-intercepts (-4 and ${ -\frac{7}{3} }$ ). This means the lines are parallel. Parallel lines, by definition, never intersect. Therefore, the system has no solution. This is a critical understanding in linear algebra and has practical implications in various modeling scenarios.

Analyzing the Statements

Now, let's evaluate the statements provided in the question:

A. The system has one solution.

As we have determined, this statement is false. The parallel nature of the lines indicates that there is no point of intersection, and hence, no single solution.

B. The system consists of parallel lines.

This statement is true. The equations have the same slope but different y-intercepts, which is the defining characteristic of parallel lines.

C. Both lines have the same slope.

This statement is also true. As we calculated, both lines have a slope of ${ \frac{1}{3} }$ .

Therefore, the two correct options are B and C.

Implications and Applications of Systems of Equations

The concept of systems of equations extends far beyond simple algebraic exercises. It is a fundamental tool in various fields:

Engineering: Engineers use systems of equations to model circuits, structures, and fluid dynamics. Determining the stability of a bridge, for example, might involve solving a complex system of equations.
Economics: Economic models often involve systems of equations to represent supply and demand, market equilibrium, and macroeconomic relationships. Analyzing these systems can help predict economic trends and inform policy decisions.
Computer Science: In computer graphics, systems of equations are used to perform transformations, such as rotations and scaling, and to render 3D objects on a 2D screen. They also play a role in optimization algorithms and machine learning.
Physics: Many physical phenomena, such as the motion of projectiles or the interaction of forces, can be described using systems of equations. Solving these systems is crucial for making predictions and understanding the underlying physics.

Understanding the nature of solutions—whether they are unique, nonexistent, or infinite—is crucial for interpreting the models and making informed decisions. For example, if a system modeling a physical process has no solution, it may indicate an inconsistency in the model or the presence of constraints that have not been accounted for.

Graphical Representation of the System

Visualizing the system of equations graphically provides an intuitive understanding of the solution. We can plot the two lines on a coordinate plane. The first line, ${ y = \frac{1}{3}x - 4 }$ , has a y-intercept of -4 and a slope of ${ \frac{1}{3} }$ . This means that for every 3 units we move to the right along the x-axis, we move 1 unit up along the y-axis.

The second line, ${ y = \frac{1}{3}x - \frac{7}{3} }$ , has a y-intercept of ${ -\frac{7}{3} }$ (approximately -2.33) and the same slope of ${ \frac{1}{3} }$ . As we plot these lines, we observe that they run parallel to each other, never intersecting. This graphical representation reinforces our algebraic conclusion that the system has no solution.

Graphing systems of equations is a powerful technique for both solving and understanding them. It provides a visual confirmation of the algebraic results and can help identify potential errors or inconsistencies in the equations.

Solving Systems of Equations Techniques

While we have determined the nature of the solution by analyzing the slopes and intercepts, there are other methods to solve systems of equations that are worth exploring:

Substitution: This method involves solving one equation for one variable and substituting that expression into the other equation. This results in a single equation with one variable, which can be solved. The solution is then substituted back into one of the original equations to find the value of the other variable.
Elimination (or Addition): This method involves manipulating the equations so that the coefficients of one variable are opposites. When the equations are added together, that variable is eliminated, leaving a single equation with one variable. This equation can be solved, and the solution is substituted back into one of the original equations to find the value of the other variable.

In our case, attempting to use substitution or elimination would lead to a contradiction, further confirming that the system has no solution. For example, if we tried to solve the first equation for y and substitute it into the second equation, we would end up with a statement that is mathematically impossible, such as 0 = a non-zero number.

Conclusion Understanding the Nature of Solutions

In conclusion, the system of equations

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

$ 3y - x = -7 $

represents two parallel lines and has no solution. This determination was made by analyzing the slopes and y-intercepts of the equations. Both lines have the same slope ( ${ \frac{1}{3} }$ ) but different y-intercepts, which is the defining characteristic of parallel lines. We also explored the graphical representation of the system, which visually confirms the lack of intersection between the lines.

Understanding the different types of solutions that systems of equations can have—one solution, no solution, or infinitely many solutions—is crucial for mathematical problem-solving and its applications in various fields. The ability to analyze systems of equations and interpret their solutions is a fundamental skill in mathematics and a valuable asset in numerous disciplines.

This comprehensive analysis demonstrates the importance of understanding the underlying principles of systems of equations and their applications. By carefully examining the equations, we were able to determine the nature of the solution and gain insights into the geometric relationship between the lines. This approach is applicable to a wide range of mathematical problems and highlights the power of mathematical reasoning and analysis.