Analyzing Systems Of Linear Equations A Comprehensive Guide

In mathematics, systems of linear equations are fundamental concepts that appear in various applications, from simple algebraic problems to complex scientific models. Understanding the properties and solutions of these systems is crucial. This article will delve into a specific system of linear equations, analyzing its characteristics and determining the correct statements about its solutions and graphical representation.

The System of Linear Equations

Let's consider the following system of linear equations:

$$\begin{aligned} 2y &= x + 10 \\ 3y &= 3x + 15 \end{aligned}$$

This system consists of two equations, each representing a straight line on a coordinate plane. To effectively analyze this system, we need to explore different approaches, including rewriting the equations in slope-intercept form, solving the system algebraically, and interpreting the graphical representation. Our goal is to determine whether the system has one solution, graphs parallel lines, or has infinitely many solutions. Through a detailed examination, we will identify the correct statements about the system.

Rewriting the Equations in Slope-Intercept Form

To better understand the nature of these equations, we will rewrite them in the slope-intercept form, which is ${ y = mx + b }$, where ${ m }$ represents the slope and ${ b }$ represents the y-intercept. This form makes it easier to visualize the lines and compare their properties.

Equation 1: ${ 2y = x + 10 }$

To rewrite this equation in slope-intercept form, we need to isolate ${ y }$ by dividing both sides of the equation by 2:

$$\begin{aligned} 2y &= x + 10 \\ \frac{2y}{2} &= \frac{x + 10}{2} \\ y &= \frac{1}{2}x + 5 \end{aligned}$$

So, the first equation in slope-intercept form is ${ y = \frac{1}{2}x + 5 }$. From this, we can see that the slope ${ m_1 }$ is ${ \frac{1}{2} }$ and the y-intercept ${ b_1 }$ is 5.

Equation 2: ${ 3y = 3x + 15 }$

Similarly, we rewrite the second equation in slope-intercept form by dividing both sides by 3:

$$\begin{aligned} 3y &= 3x + 15 \\ \frac{3y}{3} &= \frac{3x + 15}{3} \\ y &= x + 5 \end{aligned}$$

The second equation in slope-intercept form is ${ y = x + 5 }$. Here, the slope ${ m_2 }$ is 1, and the y-intercept ${ b_2 }$ is 5.

By converting the equations to slope-intercept form, we can easily compare their slopes and y-intercepts. This comparison will help us determine the relationship between the two lines and the nature of the system's solutions. The slopes and y-intercepts provide valuable information about whether the lines are parallel, intersecting, or coincident.

Analyzing the Slopes and Y-Intercepts

Now that we have both equations in slope-intercept form, we can analyze the slopes and y-intercepts to understand the relationship between the two lines. This analysis is crucial for determining whether the lines intersect at a single point, are parallel, or are the same line.

The first equation is ${ y = \frac{1}{2}x + 5 }$, with a slope ${ m_1 = \frac{1}{2} }$ and a y-intercept ${ b_1 = 5 }$. The second equation is ${ y = x + 5 }$, with a slope ${ m_2 = 1 }$ and a y-intercept ${ b_2 = 5 }$.

Comparing the Slopes

The slopes of the two lines are ${ \frac{1}{2} }$ and 1. Since the slopes are different (${ \frac{1}{2} \neq 1 }$), the lines are not parallel. Parallel lines have the same slope, so the fact that the slopes are different indicates that the lines will intersect at some point.

Comparing the Y-Intercepts

The y-intercepts of both lines are 5. This means that both lines intersect the y-axis at the same point (0, 5). However, having the same y-intercept does not necessarily mean the lines are the same; it simply means they share one common point.

Conclusion from Slope-Intercept Analysis

Since the slopes are different and the y-intercepts are the same, we can conclude that the two lines intersect at one point. This indicates that the system of equations has a unique solution. The point of intersection is the solution to the system, which satisfies both equations simultaneously. Therefore, the system has exactly one solution, and the lines are not parallel. This analysis provides a clear understanding of the graphical representation of the system and the nature of its solutions.

Solving the System Algebraically

To find the exact solution to the system, we can use algebraic methods such as substitution or elimination. These methods allow us to solve for the values of ${ x }$ and ${ y }$ that satisfy both equations simultaneously. We will use the substitution method in this case.

Using the Substitution Method

We have the following system of equations:

$$\begin{aligned} y &= \frac{1}{2}x + 5 \quad &(1) \\ y &= x + 5 \quad &(2) \end{aligned}$$

Since both equations are already solved for ${ y }$, we can set them equal to each other:

$$\frac{1}{2}x + 5 = x + 5$$

Now, we solve for ${ x }$:

$$\begin{aligned} \frac{1}{2}x + 5 &= x + 5 \\ \frac{1}{2}x &= x \\ \frac{1}{2}x - x &= 0 \\ -\frac{1}{2}x &= 0 \\ x &= 0 \end{aligned}$$

Now that we have the value of ${ x }$, we can substitute it back into either equation to find the value of ${ y }$. Let's use equation (2):

$$\begin{aligned} y &= x + 5 \\ y &= 0 + 5 \\ y &= 5 \end{aligned}$$

Thus, the solution to the system is ${ (0, 5) }$. This means the two lines intersect at the point (0, 5).

Verification

To verify our solution, we substitute the values of ${ x }$ and ${ y }$ into both original equations:

For equation (1): ${ 2y = x + 10 }$

$$\begin{aligned} 2(5) &= 0 + 10 \\ 10 &= 10 \end{aligned}$$

For equation (2): ${ 3y = 3x + 15 }$

$$\begin{aligned} 3(5) &= 3(0) + 15 \\ 15 &= 15 \end{aligned}$$

Both equations are satisfied, so our solution ${ (0, 5) }$ is correct. The algebraic solution confirms that the system has one unique solution, which is the point of intersection of the two lines.

Graphical Interpretation

The graphical representation of a system of linear equations provides a visual understanding of the solutions. Each equation represents a line, and the solution to the system is the point where the lines intersect. By graphing the equations, we can visually confirm our algebraic solution and the relationship between the lines.

Graphing the Equations

We have the equations in slope-intercept form:

$$\begin{aligned} y &= \frac{1}{2}x + 5 \\ y &= x + 5 \end{aligned}$$

The first equation, ${ y = \frac{1}{2}x + 5 }$, has a slope of ${ \frac{1}{2} }$ and a y-intercept of 5. This means the line rises 1 unit for every 2 units it runs horizontally and intersects the y-axis at the point (0, 5).

The second equation, ${ y = x + 5 }$, has a slope of 1 and a y-intercept of 5. This line rises 1 unit for every 1 unit it runs horizontally and also intersects the y-axis at the point (0, 5).

Visualizing the Intersection

When we graph these two lines, we can see that they intersect at the point (0, 5). This confirms our algebraic solution. The graphical representation clearly shows that the lines are not parallel; they intersect at a single point, indicating a unique solution to the system.

Conclusion from Graphical Analysis

The graphical analysis reinforces our previous findings. The lines intersect at one point, which corresponds to the solution ${ (0, 5) }$. This visual confirmation helps solidify our understanding of the system and its properties. The graphical representation provides an intuitive way to see the relationship between the equations and their solutions.

Determining the True Statements

Based on our analysis, we can now determine which statements about the system are true. We have analyzed the equations in slope-intercept form, solved the system algebraically, and interpreted the graphical representation. This comprehensive analysis allows us to confidently identify the correct statements.

The system of linear equations is:

$$\begin{aligned} 2y &= x + 10 \\ 3y &= 3x + 15 \end{aligned}$$

Which we rewrote in slope-intercept form as:

$$\begin{aligned} y &= \frac{1}{2}x + 5 \\ y &= x + 5 \end{aligned}$$

Evaluating the Statements

Let's consider the given statements:

A. The system has one solution.

Our algebraic solution and graphical analysis confirm that the system has one unique solution, which is ${ (0, 5) }$. The lines intersect at this point, indicating that this statement is true.

B. The system graphs parallel lines.

We determined that the slopes of the lines are ${ \frac{1}{2} }$ and 1. Since the slopes are different, the lines are not parallel. Therefore, this statement is false.

Final Conclusion

After a thorough analysis, we have determined that only statement A is true. The system has one solution, and the lines are not parallel. This conclusion is supported by our algebraic solution, graphical representation, and analysis of the slopes and y-intercepts.

Conclusion

In this article, we analyzed a system of linear equations by rewriting them in slope-intercept form, solving the system algebraically, and interpreting the graphical representation. We found that the system has one unique solution, and the lines are not parallel. This comprehensive analysis demonstrates the importance of understanding the properties of linear equations and how different methods can be used to solve and interpret them. Systems of linear equations are a fundamental topic in mathematics, and a solid understanding of these concepts is essential for further studies in algebra and beyond.

By converting the equations to slope-intercept form, we were able to compare the slopes and y-intercepts, which provided valuable information about the relationship between the lines. The algebraic solution confirmed the unique solution, and the graphical representation offered a visual confirmation of our findings. This multifaceted approach allowed us to confidently determine the true statements about the system. Understanding these concepts is crucial for solving more complex mathematical problems and real-world applications.

This detailed exploration highlights the interconnectedness of algebraic and graphical methods in analyzing linear systems. Whether dealing with simple systems or more complex ones, a thorough understanding of these principles is key to success in mathematics and related fields. The ability to analyze and solve systems of linear equations is a valuable skill that has wide-ranging applications in various disciplines, including engineering, economics, and computer science.