Solving 3x + 5 = 2x - 7 A Step-by-Step Guide
Introduction
In the realm of mathematics, solving equations is a fundamental skill. This article delves into the solution of the linear equation 3x + 5 = 2x - 7, providing a step-by-step guide and exploring the underlying concepts. Linear equations, characterized by their variable raised to the power of one, are ubiquitous in various fields, including physics, engineering, economics, and computer science. Understanding how to solve them is crucial for problem-solving and decision-making in these domains. This article will not only demonstrate the algebraic solution but also connect it to graphical interpretations, enhancing your understanding of the equation and its solutions. We will explore how the solution relates to the intersection points of lines and the x-intercepts, providing a comprehensive perspective on linear equations.
The main focus of this exploration is the equation 3x + 5 = 2x - 7. Our aim is to isolate the variable 'x' on one side of the equation to determine its value. This involves applying algebraic operations such as addition, subtraction, multiplication, and division while maintaining the equation's balance. We will break down each step, explaining the reasoning behind it and ensuring clarity in the process. Furthermore, we will discuss common pitfalls and strategies to avoid errors, making this a valuable resource for learners of all levels. Whether you are a student encountering linear equations for the first time or a seasoned mathematician seeking a refresher, this article offers a thorough and insightful guide to solving this equation.
Beyond the algebraic solution, we will explore the graphical representation of the equation. This involves interpreting the equation as a relationship between two lines and visualizing their intersection point. We will demonstrate how the x-coordinate of this intersection point corresponds to the solution of the equation. Additionally, we will touch upon the concept of x-intercepts and their relevance in the context of linear equations. By connecting the algebraic and graphical aspects, we aim to provide a holistic understanding of the equation 3x + 5 = 2x - 7 and its significance in mathematics.
Solving the Equation Algebraically
To solve the equation 3x + 5 = 2x - 7 algebraically, we need to isolate the variable 'x' on one side of the equation. This involves a series of algebraic manipulations, ensuring that we maintain the equality throughout the process. The fundamental principle behind solving equations is to perform the same operation on both sides, preserving the balance and ultimately leading to the solution.
The first step is to gather all the terms containing 'x' on one side of the equation. We can achieve this by subtracting 2x from both sides. This operation eliminates the 'x' term from the right side, leaving us with a simpler equation. The equation now becomes: 3x - 2x + 5 = 2x - 2x - 7, which simplifies to x + 5 = -7. This step is crucial as it consolidates the variable terms, making it easier to isolate 'x' in the subsequent steps.
Next, we need to isolate 'x' by eliminating the constant term on the left side. To do this, we subtract 5 from both sides of the equation. This operation cancels out the '+5' on the left side, leaving 'x' by itself. The equation now becomes: x + 5 - 5 = -7 - 5, which simplifies to x = -12. This is the solution to the equation, indicating the value of 'x' that satisfies the original equation. It's important to note that this solution is unique, as linear equations in one variable typically have only one solution.
To verify the solution, we can substitute x = -12 back into the original equation 3x + 5 = 2x - 7. This step is crucial to ensure that our algebraic manipulations were correct and that the solution we obtained is indeed valid. Substituting -12 for 'x', we get: 3(-12) + 5 = 2(-12) - 7. Simplifying both sides, we have: -36 + 5 = -24 - 7, which further simplifies to -31 = -31. Since both sides are equal, this confirms that x = -12 is the correct solution to the equation.
Graphical Interpretation: Intersection of Lines
The solution of the equation 3x + 5 = 2x - 7 can also be interpreted graphically. This involves considering each side of the equation as a separate linear function and plotting them on a coordinate plane. The point where these two lines intersect represents the solution to the equation. This graphical approach provides a visual understanding of the equation and its solution, complementing the algebraic method.
Let's consider the left side of the equation, 3x + 5, as the linear function y = 3x + 5. This represents a straight line with a slope of 3 and a y-intercept of 5. Similarly, the right side of the equation, 2x - 7, can be represented as the linear function y = 2x - 7. This is another straight line with a slope of 2 and a y-intercept of -7. By plotting these two lines on a graph, we can visually determine their intersection point.
The intersection point is the point where the two lines cross each other. At this point, the x and y coordinates satisfy both equations simultaneously. Therefore, the x-coordinate of the intersection point is the solution to the original equation 3x + 5 = 2x - 7. To find the intersection point, we can graph the two lines and visually identify the coordinates where they meet. Alternatively, we can set the two equations equal to each other, which is essentially what we did when solving the equation algebraically.
In this case, the intersection point occurs at x = -12. This is the same solution we obtained through the algebraic method. The y-coordinate of the intersection point can be found by substituting x = -12 into either of the linear equations. For instance, using y = 3x + 5, we get y = 3(-12) + 5 = -31. Therefore, the intersection point is (-12, -31). This graphical interpretation reinforces the understanding that the solution to the equation is the x-value where the two lines intersect.
Graphical Interpretation: X-intercepts and Their Relevance
While the intersection of lines provides a direct graphical interpretation of the solution, understanding x-intercepts can offer additional insights into linear equations. The x-intercept is the point where a line crosses the x-axis, meaning the y-coordinate at this point is zero. While the x-intercepts of the individual lines y = 3x + 5 and y = 2x - 7 do not directly represent the solution to the equation 3x + 5 = 2x - 7, they are important characteristics of the lines themselves and can be useful in other contexts.
To find the x-intercept of a line, we set y = 0 in the equation and solve for 'x'. For the line y = 3x + 5, setting y to zero gives us 0 = 3x + 5. Solving for 'x', we subtract 5 from both sides, resulting in -5 = 3x. Then, dividing both sides by 3, we get x = -5/3. This is the x-intercept of the line y = 3x + 5. It represents the point where the line crosses the x-axis.
Similarly, for the line y = 2x - 7, setting y to zero gives us 0 = 2x - 7. Solving for 'x', we add 7 to both sides, resulting in 7 = 2x. Then, dividing both sides by 2, we get x = 7/2. This is the x-intercept of the line y = 2x - 7. These x-intercepts are specific to each line and provide information about their position relative to the x-axis.
In the context of the equation 3x + 5 = 2x - 7, the x-intercepts of the individual lines do not directly give us the solution x = -12. Instead, the solution is found at the x-coordinate of the intersection point, as discussed earlier. However, understanding x-intercepts is crucial in various mathematical and real-world applications, such as finding the roots of a function or determining break-even points in economics. While not directly relevant to solving this particular equation, the concept of x-intercepts is a fundamental aspect of linear equations and their graphical representation.
Conclusion
In conclusion, the solution to the equation 3x + 5 = 2x - 7 is x = -12. This solution can be obtained through algebraic manipulation, as demonstrated by isolating the variable 'x' through a series of steps. We verified this solution by substituting it back into the original equation and confirming that both sides are equal. This algebraic approach is a fundamental skill in mathematics and provides a precise method for solving linear equations.
Furthermore, we explored the graphical interpretation of the equation, which offers a visual understanding of the solution. By representing each side of the equation as a linear function and plotting them on a graph, we identified the intersection point of the two lines. The x-coordinate of this intersection point, which is -12, corresponds to the solution we obtained algebraically. This graphical representation reinforces the concept that the solution to the equation is the x-value where the two lines intersect.
We also discussed the concept of x-intercepts and their relevance in the context of linear equations. While the x-intercepts of the individual lines do not directly represent the solution to the equation 3x + 5 = 2x - 7, they are important characteristics of the lines themselves. Understanding x-intercepts is crucial in various mathematical and real-world applications. By connecting the algebraic and graphical aspects, this article provides a comprehensive understanding of the equation and its solution.
Overall, solving equations is a fundamental skill in mathematics with wide-ranging applications. By mastering both algebraic and graphical techniques, you can gain a deeper understanding of equations and their solutions. The equation 3x + 5 = 2x - 7 serves as a valuable example for illustrating these concepts and techniques. Whether you are a student learning algebra or a professional applying mathematical principles, the ability to solve equations is essential for problem-solving and decision-making.
