Solving Linear Equations A Step-by-Step Guide To 5x + 7/2 = (3x/2) - 14
This article delves into the step-by-step process of solving the linear equation 5x + 7/2 = (3x/2) - 14. Linear equations, which involve variables raised to the power of one, form a foundational concept in algebra and are crucial for understanding more advanced mathematical topics. Mastering the techniques to solve these equations is essential for students and anyone working with mathematical models. In this comprehensive guide, we will break down the problem, explain each step in detail, and provide insights into the underlying principles. Understanding how to manipulate equations to isolate the variable is a fundamental skill that extends beyond mathematics, finding applications in various fields such as physics, engineering, and economics. We will explore the strategies used to simplify the equation, combine like terms, and ultimately find the value of 'x' that satisfies the equation. This methodical approach not only provides the solution but also enhances problem-solving skills applicable to a wide range of mathematical challenges. This article aims to provide a clear, concise, and thorough explanation, making it accessible to learners of all levels. By the end of this discussion, you should have a solid understanding of how to tackle similar linear equations with confidence and precision.
Step 1: Clearing the Fractions
The initial step in solving the equation 5x + 7/2 = (3x/2) - 14 involves eliminating the fractions to simplify the equation. Dealing with fractions can often complicate the solving process, so removing them makes the equation more manageable. The key to eliminating fractions is to multiply every term in the equation by the least common denominator (LCD). In this case, the only denominator present is 2, making the LCD equal to 2. By multiplying each term by 2, we ensure that the denominators will cancel out, resulting in an equation with integer coefficients. This technique is based on the fundamental property of equality, which states that performing the same operation on both sides of an equation maintains the equality. By multiplying both sides by the LCD, we maintain the balance of the equation while simplifying its structure. This step is crucial because it transforms the equation into a more familiar and easier-to-solve form. The multiplication must be applied carefully to each term, including the constants and the terms involving the variable 'x'. This ensures that the equation remains balanced and that no terms are overlooked. After this step, the equation will be free of fractions, paving the way for the subsequent steps in solving for 'x'.
To clear the fractions in the equation $5x + \frac{7}{2} = \frac{3x}{2} - 14$, we multiply every term by 2. This gives us:
Simplifying each term, we get:
Step 2: Grouping Like Terms
After eliminating the fractions, the next crucial step is to group like terms. This involves rearranging the equation so that all terms containing the variable 'x' are on one side and all constant terms are on the other side. Grouping like terms is essential for simplifying the equation and making it easier to isolate the variable. This step relies on the addition and subtraction properties of equality, which allow us to add or subtract the same value from both sides of the equation without changing its balance. The goal is to consolidate the 'x' terms and the constant terms separately, making it clear what operations need to be performed to solve for 'x'. By bringing all the 'x' terms to one side, we can combine them into a single term, and similarly, by moving all the constants to the other side, we can simplify the numerical part of the equation. This process reduces the complexity of the equation and brings us closer to isolating the variable. Careful attention to the signs of the terms is necessary when moving them across the equals sign, as adding a term on one side is equivalent to subtracting it on the other side, and vice versa. The accurate grouping of like terms is a pivotal step in solving linear equations, setting the stage for the final steps of isolating the variable and finding its value. By following this methodical approach, we ensure that the equation remains balanced and that we are progressing towards the correct solution.
To group the like terms in the equation $10x + 7 = 3x - 28$, we subtract 3x from both sides to get all the x terms on the left side:
Simplifying, we have:
Next, subtract 7 from both sides to get all the constant terms on the right side:
Which simplifies to:
Step 3: Isolating the Variable
The final step in solving the equation is isolating the variable 'x'. This means getting 'x' by itself on one side of the equation. To achieve this, we use the division property of equality, which states that dividing both sides of the equation by the same non-zero number maintains the equality. In this case, 'x' is being multiplied by 7, so we need to divide both sides of the equation by 7 to isolate 'x'. This step is crucial because it directly leads to the solution of the equation. By dividing both sides by the coefficient of 'x', we are essentially undoing the multiplication and revealing the value of 'x'. It is important to ensure that the division is performed accurately on both sides to maintain the balance of the equation and arrive at the correct solution. Once 'x' is isolated, we will have the value of 'x' that satisfies the original equation. This value can then be substituted back into the original equation to verify the solution, ensuring that the equation holds true. The process of isolating the variable is a fundamental skill in algebra, applicable to a wide range of equations and mathematical problems. By mastering this step, students can confidently solve linear equations and build a strong foundation for more advanced algebraic concepts.
To isolate the variable $x$ in the equation $7x = -35$, we divide both sides by 7:
This simplifies to:
Step 4: Verification
After finding a potential solution, the critical final step is verification. Verification involves substituting the obtained value of 'x' back into the original equation to confirm that it satisfies the equation. This process is essential because it ensures that no errors were made during the solving process and that the solution is accurate. By substituting the value, we are essentially checking if the left-hand side (LHS) of the equation is equal to the right-hand side (RHS) when 'x' is replaced with the solution. If the LHS equals the RHS, then the solution is verified and correct. If they do not match, it indicates that there was an error in the solving process, and the steps need to be reviewed to identify and correct the mistake. Verification not only confirms the solution but also reinforces the understanding of the equation and the properties of equality. It provides confidence in the answer and enhances problem-solving skills. This step is a crucial part of the mathematical process and should not be overlooked. It helps to prevent errors and ensures that the solution is mathematically sound. By consistently verifying solutions, students develop a habit of accuracy and attention to detail, which are essential for success in mathematics and other disciplines.
To verify the solution $x = -5$, substitute it back into the original equation:
Simplifying the left side:
Simplifying the right side:
Since both sides are equal, the solution $x = -5$ is correct.
Conclusion
In summary, solving the linear equation 5x + 7/2 = (3x/2) - 14 involves a series of systematic steps that, when followed carefully, lead to the correct solution. The process begins with clearing fractions by multiplying each term by the least common denominator, which simplifies the equation and makes it easier to work with. Next, grouping like terms is crucial for consolidating the 'x' terms and the constant terms separately, setting the stage for isolating the variable. Isolating the variable 'x' involves performing operations on both sides of the equation to get 'x' by itself, ultimately revealing its value. Finally, the verification step is essential to ensure the accuracy of the solution by substituting the value back into the original equation. Each of these steps is based on fundamental properties of equality, which ensure that the equation remains balanced throughout the solving process. Mastering these techniques provides a solid foundation for solving more complex equations and mathematical problems. Understanding the underlying principles and applying them methodically builds confidence and proficiency in algebra. The ability to solve linear equations is a valuable skill that extends beyond the classroom, finding applications in various real-world scenarios. By practicing and applying these steps, learners can develop a strong problem-solving approach and achieve success in mathematics.
