Solving -24x = -4: Step-by-Step Solution Explained
Have you ever encountered an equation that seemed daunting at first glance? Equations like -24x = -4 might appear tricky, but with a step-by-step approach, they become quite manageable. In this comprehensive guide, we will break down the process of solving this equation, ensuring that you not only understand the solution but also grasp the underlying principles. Whether you are a student tackling algebra or simply brushing up on your math skills, this article will provide you with the tools and knowledge to confidently solve similar equations.
Understanding the Basics of Algebraic Equations
Before we dive into the specifics of solving -24x = -4, it’s essential to understand the basic concepts of algebraic equations. An algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions involve variables (usually represented by letters like x, y, or z) and constants (numbers). The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. In simpler terms, we want to isolate the variable on one side of the equation to determine its value.
In the equation -24x = -4, we have a variable (x), a coefficient (-24), and a constant (-4). The coefficient is the number multiplied by the variable. To solve for x, we need to undo the multiplication by -24. This is achieved by using the inverse operation, which in this case is division. Understanding these foundational elements is crucial for tackling more complex equations in the future. Remember, algebra is like a puzzle; each piece (variable, constant, coefficient) fits together in a specific way, and our job is to find the missing piece.
Step-by-Step Solution to -24x = -4
Now, let’s walk through the step-by-step solution to the equation -24x = -4. This process will not only give you the answer but also help you understand the methodology behind solving such equations.
Step 1: Identify the Operation
The first step in solving any algebraic equation is to identify the operation that is being applied to the variable. In our equation, -24x = -4, the variable x is being multiplied by -24. Recognizing this is crucial because it tells us what inverse operation we need to perform.
Step 2: Apply the Inverse Operation
To isolate the variable x, we need to undo the multiplication by -24. The inverse operation of multiplication is division. Therefore, we will divide both sides of the equation by -24. It’s vital to perform the same operation on both sides of the equation to maintain the equality. This ensures that the equation remains balanced.
Step 3: Perform the Division
Dividing both sides of the equation -24x = -4 by -24, we get:
(-24x) / -24 = -4 / -24
On the left side, -24 divided by -24 cancels out, leaving us with x. On the right side, -4 divided by -24 simplifies to 1/6. Remember, a negative number divided by a negative number results in a positive number.
Step 4: Simplify the Result
After performing the division, we have:
x = 1/6
This is the solution to the equation. It means that when x is equal to 1/6, the equation -24x = -4 holds true. It’s always a good practice to simplify the result to its simplest form, which in this case, 1/6 is already in its simplest fraction form.
Verifying the Solution
Once we have a solution, it’s essential to verify it. Verifying the solution involves substituting the value we found for x back into the original equation to see if it holds true. This step ensures that we haven’t made any mistakes in our calculations and that our solution is accurate. This is a critical step in problem-solving, especially in mathematics, as it confirms the correctness of your answer. Always verify your solutions to build confidence and accuracy in your mathematical abilities.
Step 1: Substitute the Value of x
Substitute x = 1/6 back into the original equation -24x = -4:
-24 * (1/6) = -4
Step 2: Perform the Multiplication
Multiply -24 by 1/6:
-24 * (1/6) = -4
Step 3: Check for Equality
The multiplication results in -4 on the left side of the equation:
-4 = -4
Since both sides of the equation are equal, our solution x = 1/6 is correct. This verification step not only confirms the correctness of the solution but also reinforces the understanding of the equation and the solution process.
Common Mistakes to Avoid
Solving equations can be straightforward, but it’s easy to make mistakes if you’re not careful. Being aware of common pitfalls can help you avoid errors and improve your accuracy. Here are some common mistakes to watch out for:
Mistake 1: Not Applying the Operation to Both Sides
One of the most common mistakes is forgetting to apply the same operation to both sides of the equation. Remember, to maintain equality, any operation performed on one side must also be performed on the other side. For example, if you divide the left side by -24, you must also divide the right side by -24.
Mistake 2: Incorrectly Identifying the Inverse Operation
Another common mistake is using the wrong inverse operation. To undo multiplication, you must divide, and to undo addition, you must subtract. Make sure you correctly identify the operation and apply its inverse. In our equation, -24x = -4, the operation is multiplication, so the inverse operation is division.
Mistake 3: Sign Errors
Sign errors are also frequent, especially when dealing with negative numbers. Pay close attention to the signs when performing operations. A negative number divided by a negative number is positive, and a negative number multiplied by a positive number is negative. Keeping track of signs is crucial for accurate calculations.
Mistake 4: Forgetting to Simplify
Sometimes, students find the correct value for the variable but forget to simplify the result. Always simplify fractions to their lowest terms. In our case, even though 1/6 is already in its simplest form, always make it a habit to check for simplification.
Mistake 5: Skipping Verification
Skipping the verification step is a significant mistake. Verifying your solution is the best way to ensure that you have the correct answer. It only takes a few moments to substitute the value back into the original equation, and it can save you from making errors on tests or assignments.
Real-World Applications of Solving Equations
Solving equations isn’t just a skill confined to the classroom; it has numerous real-world applications. Understanding how to solve equations can be incredibly useful in various aspects of life, from managing finances to making informed decisions in your career. Equations are used to model and solve problems in fields such as engineering, physics, economics, and computer science.
For instance, in finance, equations can help you calculate interest rates, loan payments, or investment returns. In physics, equations are used to describe motion, forces, and energy. Engineers use equations to design structures, circuits, and systems. Understanding and solving equations empowers you to analyze and solve problems in a systematic way, no matter the context. The ability to solve equations is a fundamental skill that can open doors to many opportunities.
Practice Problems
To solidify your understanding of solving equations, let’s work through a few practice problems.
Practice Problem 1: 5x = 25
To solve this equation, we need to isolate x. The operation being applied to x is multiplication by 5. The inverse operation is division. Divide both sides of the equation by 5:
5x / 5 = 25 / 5
x = 5
To verify, substitute x = 5 back into the original equation:
5 * 5 = 25
25 = 25
The solution is correct.
Practice Problem 2: -3x = 12
In this equation, x is being multiplied by -3. To isolate x, we divide both sides by -3:
-3x / -3 = 12 / -3
x = -4
Verify the solution by substituting x = -4 into the original equation:
-3 * (-4) = 12
12 = 12
The solution is correct.
Practice Problem 3: 4x = -16
To solve this equation, divide both sides by 4:
4x / 4 = -16 / 4
x = -4
Verify the solution by substituting x = -4 into the original equation:
4 * (-4) = -16
-16 = -16
The solution is correct.
Conclusion
Solving equations is a fundamental skill in mathematics and a valuable tool in everyday life. By understanding the basic principles and following a systematic approach, you can confidently tackle equations like -24x = -4 and many others. Remember to identify the operation, apply the inverse operation, simplify the result, and always verify your solution. Avoiding common mistakes and practicing regularly will enhance your problem-solving skills and mathematical proficiency.
For further learning and practice, you might find helpful resources on websites like Khan Academy.
