Solve 5(x + 1.23) = 4x: A Step-by-Step Guide
Introduction
In the realm of mathematics, linear equations form the cornerstone of algebraic problem-solving. Mastering the techniques to solve these equations is crucial for various applications in science, engineering, economics, and everyday life. This article delves into the step-by-step solution of a specific linear equation: 5(x + 1.23) = 4x. We will break down the process, explaining each step in detail, ensuring a clear understanding for readers of all backgrounds. Our aim is to not only provide the solution but also to illustrate the underlying principles and strategies involved in solving linear equations. Understanding these principles will empower you to tackle a wide array of mathematical problems with confidence. Let's embark on this mathematical journey together, unraveling the solution to this equation and solidifying your grasp on linear algebra.
Understanding Linear Equations
Before we dive into the solution, let's briefly revisit the concept of linear equations. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The variables are only raised to the power of one. Graphically, a linear equation represents a straight line. Solving a linear equation means finding the value(s) of the variable(s) that make the equation true. This value, when substituted back into the equation, will satisfy the equality. In our case, the equation 5(x + 1.23) = 4x involves one variable, 'x', and our goal is to find the specific value of 'x' that makes the left-hand side of the equation equal to the right-hand side. Linear equations are fundamental because they model countless real-world situations, from calculating distances and speeds to determining financial investments and growth rates. Therefore, the ability to solve them is an indispensable skill in many fields. The process involves simplifying the equation, isolating the variable, and ultimately finding its numerical value. The beauty of linear equations lies in their simplicity and the straightforward methods used to solve them, making them an excellent starting point for anyone venturing into the world of algebra.
Step-by-Step Solution of 5(x + 1.23) = 4x
To effectively solve the equation 5(x + 1.23) = 4x, we'll follow a systematic approach, breaking it down into manageable steps. This methodical approach ensures clarity and minimizes the chances of error. Let's begin:
Step 1: Distribute the 5
The first step is to apply the distributive property. This involves multiplying the number outside the parentheses (5) by each term inside the parentheses (x and 1.23). This is a crucial step in simplifying the equation and making it easier to work with. The distributive property states that a(b + c) = ab + ac. Applying this to our equation:
5 * x + 5 * 1.23 = 4x 5x + 6.15 = 4x
Step 2: Rearrange the Equation
Now, our goal is to group the terms containing 'x' on one side of the equation and the constant terms on the other side. This is achieved by adding or subtracting terms from both sides of the equation, maintaining the equality. In our case, we'll subtract 4x from both sides to bring the 'x' terms to the left:
5x + 6.15 - 4x = 4x - 4x x + 6.15 = 0
Step 3: Isolate 'x'
To isolate 'x', we need to get it by itself on one side of the equation. We can do this by subtracting 6.15 from both sides:
x + 6.15 - 6.15 = 0 - 6.15 x = -6.15
Step 4: Solution
Therefore, the solution to the equation 5(x + 1.23) = 4x is x = -6.15. This means that if we substitute -6.15 for 'x' in the original equation, the left-hand side will equal the right-hand side.
Verification of the Solution
To ensure the accuracy of our solution, it's always a good practice to verify it. This involves substituting the calculated value of 'x' back into the original equation and checking if the equation holds true. This step helps identify any potential errors made during the solving process and boosts confidence in the final answer. Let's substitute x = -6.15 into the original equation: 5(x + 1.23) = 4x
Substituting x = -6.15
Replace 'x' with '-6.15' in the equation:
5(-6.15 + 1.23) = 4(-6.15)
Simplify the Left-Hand Side (LHS)
First, calculate the value inside the parentheses:
-6.15 + 1.23 = -4.92
Now, multiply by 5:
5 * (-4.92) = -24.6
So, the left-hand side of the equation simplifies to -24.6.
Simplify the Right-Hand Side (RHS)
Multiply 4 by -6.15:
4 * (-6.15) = -24.6
Therefore, the right-hand side of the equation also simplifies to -24.6.
Comparing LHS and RHS
We have found that:
LHS = -24.6 RHS = -24.6
Since the left-hand side equals the right-hand side (-24.6 = -24.6), our solution x = -6.15 is correct. This verification process not only confirms the solution but also reinforces the understanding of the equation and the steps taken to solve it.
Common Mistakes to Avoid
When solving linear equations, certain common mistakes can lead to incorrect answers. Being aware of these pitfalls can help you avoid them and ensure accuracy in your problem-solving. Here are some frequent errors to watch out for:
1. Incorrect Application of the Distributive Property
The distributive property, as we saw in Step 1, is crucial for simplifying equations. A common mistake is to forget to multiply the term outside the parentheses by every term inside. For example, in our equation 5(x + 1.23) = 4x, some might only multiply 5 by 'x' and forget to multiply it by 1.23. This would lead to an incorrect equation and consequently, a wrong solution. Always ensure that you distribute the term correctly to all terms within the parentheses.
2. Sign Errors
Sign errors are another frequent source of mistakes. When adding or subtracting terms from both sides of the equation, it's easy to make errors with positive and negative signs. For instance, when isolating 'x', you might add a number when you should subtract it, or vice versa. Careful attention to signs is essential to avoid these errors. Double-check each step where you manipulate signs to ensure accuracy.
3. Incorrectly Combining Like Terms
Combining like terms is a fundamental step in simplifying equations. However, mistakes can occur if you combine terms that are not alike or if you perform the addition or subtraction incorrectly. Remember that like terms are terms that have the same variable raised to the same power. Constant terms can also be combined. Ensure that you are only combining like terms and that you are doing so with the correct arithmetic operations.
4. Dividing or Multiplying by Zero
Dividing or multiplying both sides of an equation by zero is a major error. Division by zero is undefined in mathematics and will lead to nonsensical results. Multiplication by zero will make both sides of the equation equal to zero, masking the true solution. Always avoid these operations when solving equations.
5. Not Verifying the Solution
As we demonstrated earlier, verifying the solution is a crucial step in the problem-solving process. Neglecting to verify can leave you with an incorrect answer without realizing it. Substituting the solution back into the original equation and checking if it holds true is a simple yet effective way to catch mistakes and build confidence in your solution.
By being mindful of these common mistakes, you can significantly improve your accuracy and proficiency in solving linear equations. Practice and attention to detail are key to mastering these skills.
Real-World Applications of Linear Equations
Linear equations aren't just abstract mathematical concepts; they are powerful tools that have numerous real-world applications. Understanding how these equations are used in practical scenarios can make learning them more engaging and demonstrate their relevance beyond the classroom. Here are some examples of how linear equations are applied in various fields:
1. Finance
In finance, linear equations are used to calculate simple interest, loan payments, and investment growth. For instance, the formula for simple interest is a linear equation that relates the principal amount, interest rate, and time to the total interest earned. Financial analysts also use linear equations to model depreciation, cost analysis, and break-even points.
2. Physics
Physics is replete with linear relationships. Equations describing motion with constant velocity, Ohm's law relating voltage, current, and resistance, and the relationship between force, mass, and acceleration are all examples of linear equations used in physics. These equations help physicists predict and understand the behavior of physical systems.
3. Engineering
Engineers use linear equations extensively in various disciplines. Civil engineers might use them to calculate the load-bearing capacity of structures, while electrical engineers might use them to analyze circuits. Chemical engineers use linear equations in stoichiometry to balance chemical reactions and determine the amounts of reactants and products involved.
4. Economics
In economics, linear equations are used to model supply and demand curves, cost and revenue functions, and macroeconomic relationships. Economists use these equations to analyze market trends, predict economic growth, and make policy recommendations.
5. Computer Science
Linear equations are fundamental in computer graphics, linear programming, and algorithm design. They are used to perform transformations in 2D and 3D space, optimize resource allocation, and solve various computational problems.
6. Everyday Life
Linear equations also appear in everyday situations. Calculating the total cost of items with a fixed price per item, determining the distance traveled at a constant speed, and converting between different units of measurement are all examples of linear equations in action. Even budgeting and planning expenses often involve the use of linear relationships.
The prevalence of linear equations in these diverse fields highlights their importance and versatility. By mastering the techniques to solve them, you'll be equipped to tackle a wide range of practical problems.
Conclusion
In this article, we have thoroughly explored the process of solving the linear equation 5(x + 1.23) = 4x. We began by understanding the fundamental concept of linear equations and their significance in mathematics and various real-world applications. We then systematically solved the equation, breaking it down into clear, manageable steps: distributing, rearranging, and isolating the variable 'x'. Our solution, x = -6.15, was verified by substituting it back into the original equation, ensuring its accuracy. We also highlighted common mistakes to avoid when solving linear equations, such as misapplying the distributive property, sign errors, and incorrect combination of like terms. Furthermore, we discussed the wide-ranging applications of linear equations in fields like finance, physics, engineering, economics, computer science, and everyday life, emphasizing their practical relevance. By mastering the techniques and concepts presented in this article, you will be well-equipped to tackle various mathematical problems and appreciate the power and versatility of linear equations in solving real-world challenges. The ability to solve linear equations is a foundational skill that opens doors to more advanced mathematical concepts and problem-solving techniques. Keep practicing, stay attentive to detail, and you'll find that linear equations become a powerful tool in your mathematical toolkit.
