Solving -3x - 8 + 6x = 7 A Step By Step Guide

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In the realm of mathematics, solving for variables is a fundamental skill. It's the bedrock upon which more complex mathematical concepts are built. Linear equations, in particular, are a common starting point, and mastering their solution is crucial. This article will delve into the process of solving a linear equation in one variable, using the example equation -3x - 8 + 6x = 7. We will provide a step-by-step guide, ensuring clarity and understanding for learners of all levels. Furthermore, we'll discuss the importance of verifying solutions and offer insights into common pitfalls to avoid. This comprehensive approach will equip you with the tools and knowledge to confidently tackle similar equations.

Understanding Linear Equations

Before we dive into the solution, let's briefly define what a linear equation is. A linear equation in one variable is an equation that can be written in the form ax + b = 0, where 'a' and 'b' are constants, and 'x' is the variable. The key characteristic of a linear equation is that the highest power of the variable is 1. This distinguishes it from quadratic equations (where the highest power is 2) or other higher-degree equations. Linear equations represent a straight line when graphed, hence the name "linear." Solving a linear equation means finding the value of the variable that makes the equation true. This value is often referred to as the solution or the root of the equation. The process involves isolating the variable on one side of the equation by performing algebraic operations on both sides, maintaining the equality. These operations include addition, subtraction, multiplication, and division. It's crucial to perform the same operation on both sides to ensure the equation remains balanced and the solution remains valid. Understanding the underlying principles of linear equations is essential for mastering their solution and applying this knowledge to more advanced mathematical problems.

Step-by-Step Solution of -3x - 8 + 6x = 7

Now, let's break down the solution of the equation -3x - 8 + 6x = 7 into manageable steps:

Step 1: Combine Like Terms

  • Identify like terms: In this equation, the like terms are the terms involving 'x' (-3x and 6x) and the constant terms (-8 and 7). Like terms are terms that have the same variable raised to the same power. Combining like terms simplifies the equation and makes it easier to isolate the variable. This step is based on the distributive property of multiplication over addition and subtraction. By combining like terms, we are essentially reducing the number of terms in the equation, which simplifies subsequent steps. This principle is fundamental in algebra and is used extensively in simplifying expressions and solving equations. Mastering the ability to identify and combine like terms is crucial for success in algebra and beyond.

  • Combine -3x and 6x: -3x + 6x = 3x. This combines the 'x' terms into a single term. This is a straightforward application of addition of algebraic terms. We are essentially adding the coefficients of the 'x' terms while keeping the variable 'x' the same. This step is a direct application of the rules of algebra and is crucial for simplifying the equation. It allows us to reduce the number of terms involving the variable, which is a necessary step in isolating the variable and solving the equation.

  • The equation now becomes: 3x - 8 = 7. This simplified equation is much easier to work with. By combining like terms, we have transformed the original equation into a simpler form that is more amenable to further manipulation. This step is a key component of the process of solving equations, as it reduces the complexity of the equation and allows us to focus on isolating the variable. The simplified equation now contains only one term involving the variable and a constant term on each side of the equation.

Step 2: Isolate the Variable Term

  • Add 8 to both sides: To isolate the term with 'x' (3x), we need to eliminate the constant term (-8) on the left side of the equation. We do this by adding 8 to both sides of the equation. Adding the same value to both sides maintains the equality of the equation, which is a fundamental principle in algebra. This step is crucial for isolating the variable term and moving closer to solving for 'x'. The goal is to get the term involving 'x' by itself on one side of the equation.

  • (3x - 8) + 8 = 7 + 8: This shows the addition operation applied to both sides. This step clearly illustrates the algebraic manipulation being performed. Adding 8 to both sides cancels out the -8 on the left side, leaving only the term with 'x'. This is a direct application of the additive property of equality, which states that adding the same value to both sides of an equation maintains the equality.

  • This simplifies to: 3x = 15. Now, the variable term is isolated on one side. This simplified equation is a significant step towards solving for 'x'. The variable term, 3x, is now by itself on the left side of the equation, and the constant term, 15, is on the right side. This sets the stage for the final step, which involves dividing both sides by the coefficient of 'x' to isolate 'x' itself.

Step 3: Solve for x

  • Divide both sides by 3: To isolate 'x', we need to divide both sides of the equation by the coefficient of 'x', which is 3. Dividing both sides by the same non-zero value maintains the equality of the equation. This step is the final step in solving for 'x' and will give us the value of 'x' that satisfies the original equation.

  • (3x) / 3 = 15 / 3: This shows the division operation applied to both sides. This step clearly shows the algebraic operation being performed to isolate 'x'. Dividing both sides by 3 cancels out the 3 on the left side, leaving 'x' by itself. This is a direct application of the multiplicative property of equality, which states that dividing both sides of an equation by the same non-zero value maintains the equality.

  • This gives us: x = 5. Therefore, the solution to the equation -3x - 8 + 6x = 7 is x = 5. This is the value of 'x' that makes the equation true. We have successfully solved the equation by isolating 'x' using a series of algebraic manipulations. This solution can be verified by substituting 'x = 5' back into the original equation and checking if the equation holds true.

Verification of the Solution

It's always a good practice to verify your solution to ensure accuracy. To verify our solution, we substitute x = 5 back into the original equation:

  • -3(5) - 8 + 6(5) = 7
  • -15 - 8 + 30 = 7
  • 7 = 7

Since the equation holds true, our solution x = 5 is correct. Verification is a crucial step in problem-solving, especially in mathematics. It helps to catch any errors that may have been made during the solution process and provides confidence in the correctness of the answer. By substituting the solution back into the original equation, we can check if both sides of the equation are equal, which confirms the validity of the solution. This process reinforces the understanding of the equation and the meaning of its solution.

Common Mistakes to Avoid

When solving linear equations, several common mistakes can occur. Being aware of these pitfalls can help you avoid them:

  • Incorrectly combining like terms: Ensure you only combine terms with the same variable and exponent. This is a fundamental principle in algebra, and errors in combining like terms can lead to incorrect solutions. It's important to pay close attention to the signs (positive or negative) of the terms and to combine them correctly. A common mistake is to combine terms that are not like terms, such as adding a term with 'x' to a constant term.

  • Not applying operations to both sides: Remember to perform the same operation on both sides of the equation to maintain equality. This is a core principle in solving equations. Any operation performed on one side of the equation must also be performed on the other side to ensure that the equation remains balanced. Failing to do so will lead to an incorrect solution. For example, if you add a number to the left side of the equation, you must also add the same number to the right side.

  • Sign errors: Pay close attention to the signs (+ or -) when performing operations. Sign errors are a common source of mistakes in algebra. It's important to be meticulous and double-check your work to ensure that you have correctly applied the rules of sign manipulation. For example, subtracting a negative number is the same as adding a positive number, and vice versa.

  • Incorrect order of operations: Follow the order of operations (PEMDAS/BODMAS) when simplifying expressions. PEMDAS/BODMAS stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Following the correct order of operations is crucial for simplifying expressions and solving equations correctly. Failing to do so can lead to incorrect results.

Conclusion

Solving linear equations is a fundamental skill in mathematics. By following a systematic approach, such as the one outlined in this article, you can confidently solve equations like -3x - 8 + 6x = 7. Remember to combine like terms, isolate the variable term, and then solve for the variable. Always verify your solution to ensure accuracy and be mindful of common mistakes. With practice, you'll become proficient in solving linear equations and build a strong foundation for more advanced mathematical concepts. Mastering these skills not only helps in academic pursuits but also in various real-life situations where problem-solving is essential. The ability to manipulate equations and solve for unknowns is a valuable asset in many fields, from science and engineering to finance and economics.

Therefore, the correct answer is C. x = 5