Solving For X In The Equation A Step-by-Step Guide
In this article, we will delve into the process of solving for the unknown variable x in the equation $\frac{5(2x-4)}{3} + 9 = 14$. This is a common type of algebraic problem that requires a systematic approach to isolate x and find its value. Understanding how to solve such equations is a fundamental skill in mathematics, applicable in various fields from science and engineering to economics and finance. Let's break down the equation step-by-step and explore the underlying principles.
Understanding the Equation
Before we dive into the solution, let's first understand the equation itself. The equation $ rac{5(2x-4)}{3} + 9 = 14$ is a linear equation in one variable, x. Linear equations are characterized by having the variable raised to the power of 1, and they can be represented graphically as a straight line. The goal of solving a linear equation is to find the value of the variable that makes the equation true. In other words, we want to find the value of x that, when substituted into the equation, makes the left-hand side equal to the right-hand side.
The equation involves several mathematical operations: multiplication, subtraction, division, and addition. To solve for x, we need to reverse these operations in the correct order, using the principles of algebra. The key is to maintain the balance of the equation by performing the same operation on both sides. This ensures that the equation remains valid throughout the solution process.
The equation can be read as: "Five times the quantity of two times x minus four, divided by three, plus nine, equals fourteen." This verbal representation helps to understand the structure of the equation and the order of operations involved. By carefully unraveling these operations, we can isolate x and determine its value. Now, let's move on to the step-by-step solution.
Step 1: Isolate the Term with x
The first step in solving the equation $\frac{5(2x-4)}{3} + 9 = 14$ is to isolate the term that contains the variable x. This means we need to get the term $ rac{5(2x-4)}{3}$ by itself on one side of the equation. To do this, we need to eliminate the constant term, which is +9 in this case. The key here is to remember the principle of maintaining balance in the equation. Whatever operation we perform on one side, we must also perform on the other side.
Since we have +9 on the left-hand side, we can eliminate it by subtracting 9 from both sides of the equation. This is the inverse operation of addition, and it will effectively cancel out the +9 on the left. The equation then becomes:
Simplifying both sides, we get:
Now, we have successfully isolated the term containing x. The equation is now in a simpler form, making it easier to proceed with the next steps. By subtracting 9 from both sides, we have moved closer to our goal of solving for x. This step highlights the importance of using inverse operations to undo the operations in the equation and isolate the variable we are trying to find. The next step will involve dealing with the fraction and the remaining operations surrounding x. Let's continue our journey towards the solution.
Step 2: Eliminate the Fraction
Having isolated the term with x, our next task in solving the equation $\frac{5(2x-4)}{3} = 5$ is to eliminate the fraction. The fraction in this case is a division by 3, so to undo this operation, we need to multiply both sides of the equation by 3. This is another application of the principle of maintaining balance in the equation. Multiplying both sides by the same number ensures that the equation remains valid.
Multiplying both sides of the equation by 3, we get:
On the left-hand side, the multiplication by 3 cancels out the division by 3, leaving us with:
Now, the fraction is eliminated, and the equation is further simplified. We have transformed the equation into a more manageable form, which brings us closer to isolating x. This step demonstrates the power of using inverse operations to simplify equations and eliminate fractions. By multiplying both sides by the denominator of the fraction, we effectively removed the fraction from the equation, making it easier to work with. The next step will involve dealing with the parentheses and the remaining operations surrounding x. Let's proceed with the solution.
Step 3: Distribute and Simplify
With the fraction eliminated, the equation now stands as $5(2x-4) = 15$. The next step is to distribute the 5 across the terms inside the parentheses. This involves multiplying the 5 by both 2x and -4. The distributive property is a fundamental concept in algebra, which allows us to simplify expressions involving parentheses.
Applying the distributive property, we get:
Performing the multiplications, we have:
Now, the parentheses are gone, and the equation is further simplified. We have expanded the expression and combined like terms, making it easier to isolate x. This step demonstrates the importance of the distributive property in simplifying algebraic expressions. By distributing the 5 across the terms inside the parentheses, we removed the parentheses and transformed the equation into a simpler form. The next step will involve isolating the term with x and then solving for x. Let's continue our journey towards the solution.
Step 4: Isolate the x Term
Having simplified the equation to $10x - 20 = 15$, the next step is to isolate the term containing x. In this case, the term is 10x. To isolate it, we need to eliminate the constant term, which is -20. We can do this by adding 20 to both sides of the equation. This is another application of the principle of maintaining balance in the equation. Adding the same number to both sides ensures that the equation remains valid.
Adding 20 to both sides of the equation, we get:
Simplifying both sides, we have:
Now, the term with x is isolated. We have successfully eliminated the constant term from the left-hand side of the equation. This step demonstrates the importance of using inverse operations to isolate the variable we are trying to find. By adding 20 to both sides, we effectively canceled out the -20 on the left, bringing us closer to solving for x. The next step will be the final step, where we solve for x by dividing both sides by the coefficient of x. Let's proceed with the final step.
Step 5: Solve for x
With the term containing x isolated, the equation now reads $10x = 35$. The final step in solving for x is to divide both sides of the equation by the coefficient of x, which is 10 in this case. This will give us the value of x that satisfies the equation. Again, we apply the principle of maintaining balance by performing the same operation on both sides.
Dividing both sides of the equation by 10, we get:
Simplifying both sides, we find:
Therefore, the value of x that satisfies the equation $ rac{5(2x-4)}{3} + 9 = 14$ is 3.5. We have successfully solved for x by systematically applying the principles of algebra and inverse operations. This step demonstrates the final act of isolating the variable and determining its value. By dividing both sides by the coefficient of x, we effectively undid the multiplication by 10, leaving us with the value of x. Now, let's summarize the steps we took to solve this equation.
Summary of the Solution
To recap, we solved the equation $\frac{5(2x-4)}{3} + 9 = 14$ by following these steps:
- Isolate the term with x: Subtract 9 from both sides to get $\frac{5(2x-4)}{3} = 5$.
- Eliminate the fraction: Multiply both sides by 3 to get $5(2x-4) = 15$.
- Distribute and simplify: Distribute the 5 to get $10x - 20 = 15$.
- Isolate the x term: Add 20 to both sides to get $10x = 35$.
- Solve for x: Divide both sides by 10 to get $x = 3.5$.
Thus, the value of x in the equation $\frac{5(2x-4)}{3} + 9 = 14$ is 3.5. This step-by-step approach demonstrates the systematic way of solving linear equations. By carefully applying the principles of algebra and inverse operations, we can unravel the equation and isolate the variable we are trying to find. This skill is essential in mathematics and has wide applications in various fields. Now, let's discuss the importance of checking our solution.
Checking the Solution
After solving an equation, it's always a good practice to check your solution. This ensures that you haven't made any errors in your calculations and that your answer is correct. To check the solution, we substitute the value we found for x back into the original equation and see if it holds true. In this case, we found that x = 3.5, so we substitute 3.5 for x in the original equation:
Let's simplify the left-hand side of the equation:
The left-hand side of the equation equals 14, which is the same as the right-hand side. This confirms that our solution, x = 3.5, is correct. Checking the solution is a crucial step in the problem-solving process. It not only verifies the answer but also helps to reinforce the understanding of the equation and the steps involved in solving it. Now, let's consider the broader implications of solving linear equations.
The Importance of Solving Linear Equations
Solving linear equations is a fundamental skill in mathematics, and it has wide-ranging applications in various fields. Linear equations are used to model a variety of real-world phenomena, from simple relationships between quantities to more complex systems in science and engineering.
In science, linear equations are used to describe relationships between physical quantities, such as the relationship between distance, speed, and time, or the relationship between voltage, current, and resistance in an electrical circuit.
In engineering, linear equations are used in the design and analysis of structures, circuits, and systems. For example, they can be used to calculate the forces in a bridge or the flow of current in a circuit.
In economics and finance, linear equations are used to model supply and demand, calculate interest rates, and analyze financial data.
In computer science, linear equations are used in computer graphics, image processing, and machine learning.
Beyond these specific applications, the ability to solve linear equations is also a valuable problem-solving skill that can be applied in many areas of life. It teaches logical thinking, attention to detail, and the importance of following a systematic approach. By mastering the techniques for solving linear equations, you not only gain a valuable mathematical skill but also develop important problem-solving abilities that will serve you well in various aspects of life. Understanding how to manipulate equations and isolate variables is a cornerstone of mathematical literacy and a crucial skill for anyone pursuing STEM fields or any career that requires analytical thinking.
Conclusion
In conclusion, we have successfully solved the equation $\frac{5(2x-4)}{3} + 9 = 14$ and found the value of x to be 3.5. We accomplished this by following a step-by-step approach, applying the principles of algebra and inverse operations. We isolated the term with x, eliminated the fraction, distributed and simplified the equation, and finally solved for x. We also emphasized the importance of checking the solution to ensure its accuracy. This process demonstrates the systematic way of solving linear equations, a fundamental skill in mathematics with wide-ranging applications. The ability to solve linear equations is crucial for understanding and modeling real-world phenomena in various fields, from science and engineering to economics and finance. Moreover, it cultivates essential problem-solving skills, such as logical thinking, attention to detail, and the importance of a structured approach. By mastering these techniques, individuals gain a valuable tool for tackling mathematical challenges and develop a foundation for success in STEM fields and beyond. Therefore, understanding and practicing the solution of linear equations is an investment in one's mathematical proficiency and overall problem-solving capabilities.