Solving Linear Equations A Step-by-Step Guide For X - 14 = (3x + 4) / 5

by ADMIN 72 views

Introduction

Linear equations are a fundamental concept in algebra, and mastering the techniques to solve them is crucial for success in mathematics and related fields. This article focuses on providing a detailed, step-by-step guide on how to solve the linear equation x - 14 = (3_x_ + 4) / 5. We will break down each step, explaining the reasoning behind it, and ultimately arrive at the solution for x. Understanding how to solve linear equations like this one forms the bedrock for tackling more complex algebraic problems.

Understanding the Equation

Before we dive into the solution, let's first understand the structure of the equation x - 14 = (3_x_ + 4) / 5. This is a linear equation in one variable, x. Our goal is to isolate x on one side of the equation to determine its value. The equation involves a fraction, which adds a layer of complexity, but we will address this systematically. Recognizing the components of the equation—the variable, constants, and operations—is the first step in developing a strategy to solve it. This particular equation requires us to perform operations such as multiplication, addition, and subtraction while maintaining the equality of both sides.

Step 1: Eliminate the Fraction

To simplify the equation, the initial step involves eliminating the fraction. The fraction in the equation x - 14 = (3_x_ + 4) / 5 is (3_x_ + 4) / 5. To get rid of the denominator, 5, we multiply both sides of the equation by 5. This is a crucial step because it clears the fraction and makes the equation easier to work with. When we multiply both sides by 5, we get 5(x - 14) = 5((3_x_ + 4) / 5). The 5 on the right side cancels out with the denominator, leaving us with a simpler equation. This technique is a standard practice in solving equations involving fractions and helps in reducing complexity.

Step 2: Distribute on the Left Side

After eliminating the fraction, the next step is to distribute the 5 on the left side of the equation. Starting from the transformed equation 5(x - 14) = 3_x_ + 4, we distribute the 5 across the terms inside the parentheses. This means we multiply both x and -14 by 5. This distribution results in 5*x* - 70. The equation now looks like 5*x* - 70 = 3*x* + 4. Distributing correctly is vital for maintaining the equality and accurately progressing towards the solution. This step prepares the equation for further simplification by grouping like terms.

Step 3: Group Like Terms

Now that we've distributed and simplified, the next crucial step is to group the like terms. We have the equation 5*x* - 70 = 3*x* + 4. Our goal is to bring all the x terms to one side and all the constant terms to the other side. To do this, we can subtract 3*x* from both sides of the equation. This gives us 5*x* - 3*x* - 70 = 3*x* - 3*x* + 4, which simplifies to 2*x* - 70 = 4. Grouping like terms is a fundamental technique in algebra, making the equation easier to solve by isolating the variable term. This step streamlines the equation towards the final solution.

Step 4: Isolate the Variable Term

Following the grouping of like terms, the next step is to isolate the variable term. Currently, we have the equation 2*x* - 70 = 4. To isolate the term with x (which is 2*x), we need to get rid of the constant term on the same side, which is -70. We can do this by adding 70 to both sides of the equation. This maintains the balance of the equation and moves us closer to solving for x. Adding 70 to both sides gives us 2x* - 70 + 70 = 4 + 70, which simplifies to 2*x* = 74. Isolating the variable term is a key strategy in solving equations, setting up the final step of finding the value of x.

Step 5: Solve for x

The final step in solving the equation is to find the value of x. We have arrived at the equation 2*x* = 74. To solve for x, we need to isolate x completely. This can be achieved by dividing both sides of the equation by the coefficient of x, which is 2. Dividing both sides by 2 gives us (2*x) / 2 = 74 / 2. This simplifies to x = 37. Therefore, the solution to the equation x - 14 = (3x* + 4) / 5 is x = 37. This final step demonstrates the culmination of all previous steps, leading to the determination of the variable's value.

Verification

To ensure our solution is correct, we can substitute x = 37 back into the original equation and check if both sides are equal. The original equation is x - 14 = (3*x* + 4) / 5. Substituting x = 37, we get 37 - 14 = (3(37) + 4) / 5. Simplifying the left side gives us 23. On the right side, 3(37) + 4 equals 111 + 4, which is 115. Then, 115 / 5 equals 23. Since both sides of the equation equal 23, our solution x = 37 is correct. Verification is an essential practice in solving equations to confirm the accuracy of the solution.

Conclusion

In this article, we've provided a comprehensive guide on how to solve the linear equation x - 14 = (3*x* + 4) / 5. By following the steps—eliminating the fraction, distributing, grouping like terms, isolating the variable term, and solving for x—we successfully found that x = 37. Remember, each step is crucial for arriving at the correct answer. Furthermore, verifying the solution by substituting it back into the original equation is an excellent way to confirm its accuracy. Mastering these techniques will empower you to confidently tackle various algebraic problems. Practice is key to building proficiency, so continue to apply these methods to other linear equations.