Solving 4(x+2)-13(2x-3)=158 A Step-by-Step Guide

Introduction

In this detailed guide, we will walk through the process of solving the linear equation 4(x+2)-13(2x-3)=158. Linear equations are a fundamental concept in algebra, and mastering their solution is crucial for further studies in mathematics and related fields. This article aims to provide a clear, step-by-step explanation of how to solve this specific equation, while also reinforcing the general principles applicable to solving any linear equation. Whether you are a student learning algebra for the first time, or someone looking to refresh your skills, this guide will provide you with the tools and understanding needed to tackle such problems with confidence. The method we will use involves expanding the equation by applying the distributive property, combining like terms, and isolating the variable x to find its value. This approach is standard and effective for solving a wide range of linear equations. Let’s delve into the specifics of the equation at hand, and by the end of this guide, you will have a solid understanding of how to solve it.

Understanding Linear Equations

Before diving into the specifics of the equation 4(x+2)-13(2x-3)=158, it's essential to understand what linear equations are and the basic principles involved in solving them. 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. These equations are called “linear” because when plotted on a graph, they form a straight line. The general form of a linear equation in one variable is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we want to solve for. The key to solving linear equations lies in isolating the variable on one side of the equation. This is achieved by performing the same operations on both sides of the equation, maintaining the equality. These operations typically include addition, subtraction, multiplication, and division. For example, if we have an equation like 2x + 3 = 7, we would first subtract 3 from both sides to get 2x = 4, and then divide both sides by 2 to find x = 2. This principle of maintaining balance by performing identical operations on both sides is fundamental. When dealing with more complex linear equations, such as the one we are addressing, we often need to simplify the equation first. This involves expanding expressions using the distributive property and combining like terms. The distributive property states that a(b + c) = ab + ac, which is crucial for handling expressions like 4(x+2) and -13(2x-3) in our equation. By understanding these basic principles, you will be well-equipped to solve a variety of linear equations, including the one we are focusing on.

Step-by-Step Solution of 4(x+2)-13(2x-3)=158

Now, let’s break down the solution to the equation 4(x+2)-13(2x-3)=158 step-by-step. This will provide a clear understanding of the process and the reasoning behind each step. The first crucial step in solving this equation is to apply the distributive property to remove the parentheses. This involves multiplying the constants outside the parentheses by each term inside. For the first term, 4(x+2), we multiply 4 by x and 4 by 2, resulting in 4x + 8. Similarly, for the second term, -13(2x-3), we multiply -13 by 2x and -13 by -3, resulting in -26x + 39. Note the importance of paying attention to signs, as multiplying a negative number by a negative number yields a positive number. After applying the distributive property, our equation becomes 4x + 8 - 26x + 39 = 158. The next step is to combine like terms on the left side of the equation. Like terms are terms that contain the same variable raised to the same power. In our equation, 4x and -26x are like terms, as are the constants 8 and 39. Combining these, we add 4x and -26x to get -22x, and we add 8 and 39 to get 47. This simplifies our equation to -22x + 47 = 158. Now, we need to isolate the term with the variable, which is -22x. To do this, we subtract 47 from both sides of the equation. This maintains the balance of the equation and moves us closer to isolating x. Subtracting 47 from both sides gives us -22x = 158 - 47, which simplifies to -22x = 111. Finally, to solve for x, we divide both sides of the equation by the coefficient of x, which is -22. This gives us x = 111 / -22. Performing the division, we find that x = -5.045 (approximately). This is the solution to our linear equation. Each step in this process is vital, from applying the distributive property to combining like terms and finally isolating the variable. Understanding and mastering these steps will enable you to solve a wide variety of linear equations.

Detailed Breakdown of Each Step

To further clarify the solution process, let’s delve into a more detailed breakdown of each step involved in solving the equation 4(x+2)-13(2x-3)=158. This will provide a deeper understanding of the underlying principles and techniques used. Step 1: Applying the Distributive Property The distributive property is a fundamental concept in algebra that allows us to simplify expressions involving parentheses. It states that a(b + c) = ab + ac. In our equation, we have two terms with parentheses: 4(x+2) and -13(2x-3). Applying the distributive property to the first term, 4(x+2), we multiply 4 by both x and 2. This gives us 4 * x + 4 * 2, which simplifies to 4x + 8. For the second term, -13(2x-3), we multiply -13 by both 2x and -3. This gives us -13 * 2x + (-13) * (-3), which simplifies to -26x + 39. It is crucial to pay attention to the signs here, as multiplying a negative number by a negative number results in a positive number. After applying the distributive property, our equation becomes 4x + 8 - 26x + 39 = 158. This step is essential for removing the parentheses and preparing the equation for further simplification. Step 2: Combining Like Terms After applying the distributive property, we need to combine like terms to simplify the equation further. Like terms are terms that have the same variable raised to the same power. In our equation, the like terms are 4x and -26x, and the constants 8 and 39. To combine the variable terms, we add 4x and -26x, which results in -22x. To combine the constants, we add 8 and 39, which results in 47. After combining like terms, our equation simplifies to -22x + 47 = 158. This step reduces the complexity of the equation, making it easier to solve. Step 3: Isolating the Variable Term Our goal is to isolate the variable x on one side of the equation. To do this, we first need to isolate the term containing x, which is -22x. In our equation, -22x + 47 = 158, we can isolate -22x by subtracting 47 from both sides of the equation. This is based on the principle that performing the same operation on both sides of an equation maintains the equality. Subtracting 47 from both sides gives us -22x = 158 - 47, which simplifies to -22x = 111. This step brings us closer to isolating x. Step 4: Solving for x The final step is to solve for x by dividing both sides of the equation by the coefficient of x, which is -22. This gives us x = 111 / -22. Performing the division, we find that x ≈ -5.045. This is the solution to the equation. By breaking down each step in detail, we can see the logical progression and the application of algebraic principles that lead to the solution. This comprehensive approach ensures a clear understanding of the process and the underlying concepts.

Common Mistakes to Avoid

When solving linear equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure accurate solutions. One of the most frequent errors is related to the distributive property. As we saw in the step-by-step solution, it’s crucial to correctly multiply the constant outside the parentheses by each term inside. A common mistake is forgetting to distribute the constant to all terms, or incorrectly handling the signs. For instance, in the equation 4(x+2)-13(2x-3)=158, failing to distribute the -13 correctly can lead to an incorrect equation. Another common mistake occurs when combining like terms. Students may incorrectly combine terms that are not like terms, such as adding a term with x to a constant. Remember, like terms must have the same variable raised to the same power. Additionally, errors in arithmetic can easily occur when adding or subtracting numbers, especially when dealing with negative numbers. Double-checking your calculations can help prevent these mistakes. Another area where errors often arise is when isolating the variable. It's essential to perform the same operation on both sides of the equation to maintain the balance. Forgetting to do so, or performing the wrong operation, will lead to an incorrect solution. For example, if you have -22x + 47 = 158, you must subtract 47 from both sides before dividing by -22. A final common mistake is not paying attention to the order of operations. Remember to follow the PEMDAS/BODMAS rule (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This ensures that you perform operations in the correct sequence. To avoid these mistakes, it’s helpful to write out each step clearly and double-check your work. Practice is also key to mastering the process of solving linear equations and avoiding common errors.

Real-World Applications of Linear Equations

Linear equations are not just abstract mathematical concepts; they have numerous practical applications in real-world scenarios. Understanding these applications can help you appreciate the importance of mastering linear equations. One common application is in financial planning. For example, linear equations can be used to calculate simple interest on a loan or investment. The formula for simple interest, I = PRT (where I is the interest, P is the principal, R is the interest rate, and T is the time), is a linear equation. Similarly, budgeting often involves linear equations, where expenses and income can be modeled and analyzed to make financial decisions. In physics, linear equations are used to describe motion with constant velocity. The equation d = vt (where d is the distance, v is the velocity, and t is the time) is a linear equation that can help calculate how far an object travels in a given time at a constant speed. In engineering, linear equations are used in circuit analysis, structural analysis, and many other areas. For example, Ohm's Law (V = IR, where V is voltage, I is current, and R is resistance) is a linear equation used in electrical engineering. Chemistry also utilizes linear equations, such as in stoichiometry, where equations are used to balance chemical reactions and calculate the amounts of reactants and products. Everyday life is also full of situations where linear equations come into play. For instance, calculating the total cost of items at a store, determining the amount of time it will take to travel a certain distance at a constant speed, or even adjusting a recipe to serve a different number of people can all involve linear equations. By recognizing the widespread applications of linear equations, you can see their importance in various fields and everyday situations. This understanding can motivate you to master the skills needed to solve them effectively.

Conclusion

In conclusion, solving the linear equation 4(x+2)-13(2x-3)=158 involves a systematic approach that includes applying the distributive property, combining like terms, and isolating the variable. This step-by-step process not only helps in solving this specific equation but also provides a framework for tackling a wide range of linear equations. Understanding the underlying principles, such as the distributive property and the importance of maintaining balance in an equation, is crucial for success in algebra and beyond. By avoiding common mistakes, such as incorrect distribution or arithmetic errors, you can improve your accuracy and confidence in solving linear equations. Furthermore, recognizing the real-world applications of linear equations in fields like finance, physics, engineering, and everyday situations underscores their practical importance. Mastering these skills is essential for academic pursuits and can be beneficial in various aspects of life. We encourage you to practice solving linear equations regularly to reinforce your understanding and develop proficiency. With consistent effort and a clear understanding of the concepts, you can confidently solve linear equations and apply them to real-world problems.