Analyzing Slopes And Y-Intercepts Of The System Of Equations 4x + 2y = -2 And X - 3y = 24

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In the realm of mathematics, systems of equations serve as powerful tools for modeling and solving real-world problems. These systems often involve two or more equations with multiple variables, and their solutions represent the points where the equations intersect. A crucial aspect of understanding systems of equations lies in analyzing the slopes and y-intercepts of the lines they represent. In this article, we will delve into a specific system of equations and dissect its characteristics, focusing on how the slopes and y-intercepts dictate the behavior and solutions of the system.

Deconstructing the System of Equations

To begin our exploration, let's consider the following system of equations:

4x + 2y = -2
x - 3y = 24

These two linear equations represent straight lines when plotted on a coordinate plane. To gain a deeper understanding of their relationship, we need to transform them into slope-intercept form, which is expressed as:

y = mx + b

where 'm' represents the slope of the line and 'b' represents the y-intercept (the point where the line crosses the y-axis). By converting the equations to slope-intercept form, we can directly compare their slopes and y-intercepts, revealing valuable insights into the system's behavior.

Transforming the Equations

Let's start by transforming the first equation, 4x + 2y = -2, into slope-intercept form:

  1. Subtract 4x from both sides: 2y = -4x - 2
  2. Divide both sides by 2: y = -2x - 1

Now, let's transform the second equation, x - 3y = 24, into slope-intercept form:

  1. Subtract x from both sides: -3y = -x + 24
  2. Divide both sides by -3: y = (1/3)x - 8

Now that both equations are in slope-intercept form, we can readily identify their slopes and y-intercepts.

Dissecting the Slopes and Y-Intercepts

From the transformed equations, we can extract the following information:

  • Equation 1: y = -2x - 1
    • Slope (m1) = -2
    • Y-intercept (b1) = -1
  • Equation 2: y = (1/3)x - 8
    • Slope (m2) = 1/3
    • Y-intercept (b2) = -8

By comparing the slopes and y-intercepts, we can draw significant conclusions about the relationship between the two lines.

Slope Analysis

The slopes of the two lines are different (m1 = -2 and m2 = 1/3). This crucial observation tells us that the lines are not parallel. Parallel lines have the same slope and never intersect. Since the slopes are different, the lines will intersect at a single point, which represents the solution to the system of equations. Furthermore, the slopes have opposite signs (one negative and one positive). This indicates that the lines are not only non-parallel but also intersect at an angle other than 90 degrees. If the slopes were negative reciprocals of each other (e.g., -2 and 1/2), the lines would be perpendicular, intersecting at a 90-degree angle.

The different slopes also imply that the system of equations is independent. An independent system has a unique solution, meaning there is only one point (x, y) that satisfies both equations simultaneously. This unique solution is the point of intersection of the two lines on the graph.

Y-Intercept Analysis

The y-intercepts of the two lines are also different (b1 = -1 and b2 = -8). The y-intercept is the point where the line crosses the y-axis. Since the y-intercepts are different, the lines intersect the y-axis at distinct points. This further confirms that the lines are not the same line, as identical lines would have the same y-intercept.

The y-intercepts provide additional information about the position of the lines on the coordinate plane. The first line intersects the y-axis at -1, while the second line intersects the y-axis at -8. This visual separation reinforces the fact that the lines are distinct and will intersect at a single point.

Graphical Interpretation

The graph of the system of equations visually reinforces our analysis of the slopes and y-intercepts. When we plot the two lines on a coordinate plane, we observe that they intersect at a single point. This point of intersection represents the solution to the system of equations, the values of x and y that satisfy both equations simultaneously.

The graphical representation provides a clear picture of the relationship between the lines. The different slopes are evident in the distinct angles at which the lines rise or fall. The different y-intercepts are also visually apparent as the points where the lines cross the y-axis. The point of intersection, the solution to the system, is the unique point where the two lines meet.

By analyzing the graph in conjunction with the slopes and y-intercepts, we gain a comprehensive understanding of the system of equations and its solution.

Determining the Solution

To find the exact solution to the system of equations, we can use various methods, such as substitution or elimination. These algebraic techniques allow us to solve for the values of x and y that satisfy both equations. The solution we obtain algebraically will correspond to the point of intersection we observed graphically.

Solving by Substitution

Let's use the substitution method to solve the system:

  1. Solve the second equation for x: x = 3y + 24
  2. Substitute this expression for x into the first equation: 4(3y + 24) + 2y = -2
  3. Simplify and solve for y: 12y + 96 + 2y = -2 => 14y = -98 => y = -7
  4. Substitute the value of y back into the equation x = 3y + 24: x = 3(-7) + 24 => x = 3

Therefore, the solution to the system of equations is x = 3 and y = -7, or the point (3, -7).

Solving by Elimination

Alternatively, we can use the elimination method:

  1. Multiply the second equation by -4: -4x + 12y = -96
  2. Add the modified second equation to the first equation: (4x + 2y) + (-4x + 12y) = -2 + (-96)
  3. Simplify and solve for y: 14y = -98 => y = -7
  4. Substitute the value of y back into either of the original equations to solve for x. Let's use the first equation: 4x + 2(-7) = -2 => 4x - 14 = -2 => 4x = 12 => x = 3

Again, we find the solution to be x = 3 and y = -7, or the point (3, -7).

Both methods confirm that the solution to the system of equations is (3, -7), which is the point where the two lines intersect on the graph.

Conclusion: A Holistic Understanding

In conclusion, by analyzing the slopes and y-intercepts of the system of equations, we have gained a comprehensive understanding of the relationship between the two lines. The different slopes indicate that the lines are not parallel and will intersect at a single point, signifying a unique solution to the system. The different y-intercepts further confirm that the lines are distinct and intersect the y-axis at different points.

The slopes and y-intercepts are fundamental properties of linear equations that provide valuable insights into their behavior and relationships. By understanding these concepts, we can effectively analyze and solve systems of equations, gaining a deeper appreciation for the power of mathematics in modeling and solving real-world problems.

The graphical representation of the system reinforces our analysis, visually demonstrating the intersection of the lines at the solution point. By combining algebraic and graphical techniques, we can confidently determine the solution and gain a holistic understanding of the system of equations.

This analysis highlights the importance of slopes and y-intercepts in understanding the behavior of linear equations and systems of equations. By mastering these concepts, we can unlock the power of mathematics to solve a wide range of problems.

Ultimately, the examination of this system of equations underscores the interconnectedness of algebraic and geometric concepts. Understanding the slopes and y-intercepts allows us to predict the graphical behavior of the lines, and conversely, the graphical representation provides a visual confirmation of our algebraic analysis. This dual perspective is crucial for developing a deep and intuitive understanding of mathematics.