Solving -3x - 4 = -15 A Step-by-Step Guide

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In the realm of mathematics, linear equations form the bedrock of algebraic problem-solving. Mastering the art of solving these equations is crucial for success in higher-level math and various real-world applications. This article delves into the step-by-step process of solving linear equations, focusing on the fundamental principle of isolating the variable. We will dissect the equation -3x - 4 = -15, providing a detailed walkthrough to arrive at the solution. By understanding the underlying concepts and techniques, you'll gain the confidence to tackle a wide range of linear equations with ease.

Understanding Linear Equations

Before diving into the solution, let's establish a clear understanding of what linear equations are. 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 involve only basic arithmetic operations: addition, subtraction, multiplication, and division. The variable in a linear equation is raised to the power of one, and the equation, when graphed, produces a straight line. The general form of a linear equation is ax + b = c, where 'a', 'b', and 'c' are constants, and 'x' is the variable we aim to solve for. Understanding this basic structure is essential for approaching and solving any linear equation effectively.

The Core Principle: Isolating the Variable

The key to solving linear equations lies in a simple yet powerful principle: isolating the variable. This means manipulating the equation using algebraic operations to get the variable (in our case, 'x') alone on one side of the equation. To achieve this, we employ inverse operations, which are operations that undo each other. For instance, addition is the inverse operation of subtraction, and multiplication is the inverse operation of division. By strategically applying inverse operations, we systematically eliminate terms surrounding the variable until it stands alone, revealing its value. This process requires careful attention to maintaining the balance of the equation, ensuring that any operation performed on one side is also performed on the other.

Solving -3x - 4 = -15: A Step-by-Step Solution

Now, let's apply the principle of isolating the variable to solve the equation -3x - 4 = -15. We'll break down the process into clear, manageable steps, explaining the reasoning behind each action.

Step 1: Isolate the Term with the Variable

Our initial goal is to isolate the term containing the variable, which is -3x. To do this, we need to eliminate the constant term, -4, on the left side of the equation. The inverse operation of subtraction is addition, so we add 4 to both sides of the equation. This maintains the equality and moves us closer to isolating the variable.

-3x - 4 + 4 = -15 + 4

This simplifies to:

-3x = -11

Step 2: Isolate the Variable

Now that we have -3x = -11, the variable 'x' is being multiplied by -3. To isolate 'x', we need to undo this multiplication. The inverse operation of multiplication is division, so we divide both sides of the equation by -3.

(-3x) / -3 = (-11) / -3

This simplifies to:

x = 11/3

Step 3: Express the Solution

Therefore, the solution to the equation -3x - 4 = -15 is x = 11/3. This is a reduced fraction, as requested. We have successfully isolated the variable and found its value. This solution can be verified by substituting 11/3 back into the original equation to ensure that both sides are equal.

Verifying the Solution

To ensure the accuracy of our solution, it's always a good practice to verify it. We do this by substituting the calculated value of 'x' back into the original equation and checking if it holds true. Let's substitute x = 11/3 into the equation -3x - 4 = -15:

-3 * (11/3) - 4 = -15

First, we multiply -3 by 11/3:

-11 - 4 = -15

Then, we perform the subtraction:

-15 = -15

Since both sides of the equation are equal, our solution x = 11/3 is correct. Verification is a crucial step in the problem-solving process, as it helps identify any potential errors and reinforces the understanding of the solution.

Common Mistakes and How to Avoid Them

Solving linear equations can sometimes be tricky, and it's easy to make mistakes if you're not careful. Here are some common errors and tips on how to avoid them:

  1. Incorrectly Applying Inverse Operations: A frequent mistake is using the wrong inverse operation or applying it only to one side of the equation. Remember to always perform the same operation on both sides to maintain balance and use the correct inverse operation (addition for subtraction, multiplication for division, and vice versa).

  2. Sign Errors: Pay close attention to the signs (+ and -) of the numbers. A simple sign error can lead to an incorrect solution. Double-check your work, especially when dealing with negative numbers.

  3. Order of Operations: Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Follow this order to correctly simplify expressions.

  4. Combining Unlike Terms: You can only combine like terms (terms with the same variable and exponent). For example, you can combine 3x and 5x, but you cannot combine 3x and 5. Make sure you're combining the correct terms.

  5. Forgetting to Distribute: If you have an expression like 2(x + 3), remember to distribute the 2 to both terms inside the parentheses: 2 * x + 2 * 3. Neglecting to distribute can lead to errors.

By being aware of these common mistakes and practicing regularly, you can improve your accuracy and confidence in solving linear equations.

Practice Problems

To solidify your understanding, let's try a few more practice problems:

  1. Solve for y: 5y + 7 = 22

  2. Solve for z: -2z - 9 = 1

  3. Solve for a: 4a + 3 = -5

  4. Solve for b: -6b + 10 = -2

Work through these problems using the steps outlined earlier. Remember to isolate the variable and verify your solutions. The more you practice, the more comfortable and proficient you'll become at solving linear equations.

Real-World Applications of Linear Equations

Linear equations aren't just abstract mathematical concepts; they have numerous real-world applications. They are used in various fields, including:

  • Finance: Calculating interest, loans, and investments.
  • Physics: Describing motion, forces, and energy.
  • Engineering: Designing structures, circuits, and systems.
  • Economics: Modeling supply and demand, and predicting economic trends.
  • Computer Science: Developing algorithms and solving computational problems.

Understanding linear equations can help you make informed decisions in your personal and professional life. For example, you can use them to calculate the cost of a loan, determine the distance traveled at a constant speed, or predict the growth of a population. The ability to solve linear equations is a valuable skill that can empower you to analyze and solve problems in a variety of contexts.

Conclusion

In conclusion, solving linear equations is a fundamental skill in mathematics with wide-ranging applications. By mastering the principle of isolating the variable and following a step-by-step approach, you can confidently solve a variety of linear equations. Remember to verify your solutions and be mindful of common mistakes. With practice and perseverance, you'll develop a strong foundation in algebra and enhance your problem-solving abilities.