Solving Linear Equations A Step-by-Step Guide With Example

In the realm of mathematics, solving linear equations is a fundamental skill. Linear equations, characterized by variables raised to the power of one, appear in various applications, from basic algebra to advanced calculus. This article delves into the process of solving a specific linear equation, 3(x-3) + 13 = 2(x+5), providing a step-by-step guide and emphasizing the importance of checking the solution. Understanding how to solve these equations is crucial for success in many areas of mathematics and related fields. We will break down each step, ensuring clarity and providing insights into the underlying principles. This will not only help in solving this particular equation but also build a solid foundation for tackling more complex problems.

Unveiling the Linear Equation: 3(x-3) + 13 = 2(x+5)

The equation we aim to solve is 3(x-3) + 13 = 2(x+5). This equation involves the variable 'x', and our goal is to isolate 'x' on one side of the equation to determine its value. The equation's structure involves parentheses, which indicate the need for the distributive property, a key concept in simplifying algebraic expressions. The presence of constants and coefficients adds to the equation's complexity, requiring careful application of algebraic principles to arrive at the correct solution. The process involves a series of steps, each designed to simplify the equation while maintaining its balance. These steps include distributing constants over parentheses, combining like terms, and isolating the variable. By mastering these techniques, you will gain confidence in handling linear equations of various forms.

Step 1: Applying the Distributive Property

The first step in solving the equation involves applying the distributive property. This property states that a(b + c) = ab + ac. Applying this to our equation, we distribute the 3 across (x-3) and the 2 across (x+5):

3(x-3) becomes 3 * x - 3 * 3 = 3x - 9 2(x+5) becomes 2 * x + 2 * 5 = 2x + 10

Substituting these back into the equation, we get:

3x - 9 + 13 = 2x + 10

The distributive property is a fundamental concept in algebra and is essential for simplifying expressions and solving equations. It allows us to remove parentheses, making the equation easier to manipulate. Understanding this property is crucial for mastering algebraic manipulations and solving a wide range of mathematical problems. The ability to correctly apply the distributive property is a cornerstone of algebraic proficiency.

Step 2: Combining Like Terms

The next step involves combining like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. In our equation, we have constant terms (-9 and +13) on the left side. Combining these, we get:

-9 + 13 = 4

So, the equation becomes:

3x + 4 = 2x + 10

Combining like terms simplifies the equation by reducing the number of terms, making it easier to isolate the variable. This step is crucial for maintaining the equation's balance and ensuring that the solution remains accurate. By combining like terms, we are essentially tidying up the equation, making it more manageable and setting the stage for the next steps in the solution process.

Step 3: Isolating the Variable

To isolate the variable 'x', we need to get all the 'x' terms on one side of the equation and the constant terms on the other side. We can do this by subtracting 2x from both sides:

3x + 4 - 2x = 2x + 10 - 2x

This simplifies to:

x + 4 = 10

Now, we subtract 4 from both sides to isolate 'x':

x + 4 - 4 = 10 - 4

This gives us:

x = 6

Isolating the variable is a key step in solving any equation. It involves performing operations on both sides of the equation to separate the variable from the constants and coefficients. This process requires careful attention to maintain the equation's balance and ensure that the solution remains valid. By isolating the variable, we effectively determine the value of the unknown, which is the ultimate goal of solving the equation.

Checking the Solution: Ensuring Accuracy

After finding a solution, it's essential to check its accuracy. This involves substituting the solution back into the original equation to see if it holds true. This step helps to identify any errors made during the solving process and ensures that the solution is correct. Verification is a crucial aspect of problem-solving in mathematics, as it provides confidence in the accuracy of the result.

Step 4: Substituting the Solution

Substitute x = 6 back into the original equation:

3(x-3) + 13 = 2(x+5) 3(6-3) + 13 = 2(6+5)

Step 5: Evaluating Both Sides

Evaluate both sides of the equation:

3(3) + 13 = 2(11) 9 + 13 = 22 22 = 22

Since both sides of the equation are equal, the solution x = 6 is correct. Checking the solution is a critical step in the problem-solving process. It helps to catch any errors that may have occurred during the algebraic manipulations and ensures that the final answer is accurate. This step reinforces the understanding of the equation and the solution, providing a sense of confidence in the result.

Conclusion: The Solution and Its Significance

The solution to the linear equation 3(x-3) + 13 = 2(x+5) is x = 6. This solution satisfies the equation, as verified by substituting it back into the original equation. The process of solving linear equations involves applying algebraic principles such as the distributive property, combining like terms, and isolating the variable. Each step is crucial in arriving at the correct solution. Furthermore, checking the solution is essential to ensure accuracy and validate the result.

Understanding how to solve linear equations is a fundamental skill in mathematics. It forms the basis for solving more complex equations and is applied in various fields, including science, engineering, and economics. By mastering the techniques discussed in this article, you will be well-equipped to tackle a wide range of mathematical problems.

The solution set is {6}.

This detailed explanation not only provides the solution but also emphasizes the underlying concepts and steps involved in solving linear equations. This comprehensive approach enhances understanding and builds a strong foundation for future mathematical endeavors.