Solve The Equation -42x - 5 + 6 = 10
In this article, we will walk through the steps to solve the linear equation -42x - 5 + 6 = 10. Linear equations are a fundamental concept in algebra, and understanding how to solve them is crucial for more advanced mathematical topics. We will break down the problem step-by-step, ensuring a clear and concise explanation for readers of all levels. This comprehensive guide will cover simplification, isolation of the variable, and verification of the solution.
Understanding Linear Equations
Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable raised to the power of one. The general form of a linear equation in one variable is ax + b = c, where x is the variable, and a, b, and c are constants. Solving a linear equation means finding the value of the variable that makes the equation true. This involves isolating the variable on one side of the equation by performing the same operations on both sides to maintain equality.
To effectively tackle linear equations, it is essential to grasp the underlying principles of algebraic manipulation. The key concept is maintaining balance – whatever operation you perform on one side of the equation, you must also perform on the other side. This ensures that the equality remains valid. Common operations include addition, subtraction, multiplication, and division. Additionally, simplifying the equation by combining like terms is a crucial step in the problem-solving process. Like terms are terms that have the same variable raised to the same power, allowing us to consolidate the equation and make it easier to solve.
For instance, in the equation -42x - 5 + 6 = 10, we have terms involving the variable x and constant terms. Our goal is to isolate x on one side of the equation. Before we can isolate x, we need to simplify the equation by combining the constant terms. This involves adding or subtracting the constants to create a single constant term on the left side of the equation. After simplification, we can proceed to isolate x by performing inverse operations. Inverse operations are operations that undo each other – for example, addition and subtraction are inverse operations, as are multiplication and division. By strategically applying inverse operations, we can systematically isolate the variable and determine its value. This meticulous approach ensures accuracy and efficiency in solving linear equations.
Step-by-Step Solution
To solve the given equation, -42x - 5 + 6 = 10, we will follow these steps:
Step 1: Simplify the Equation
First, we need to simplify the equation by combining like terms. In this case, we have two constant terms on the left side: -5 and +6. Combining these gives us:
-42x + (-5 + 6) = 10
-42x + 1 = 10
Simplifying an equation is a crucial initial step because it reduces the complexity and makes subsequent steps more manageable. By combining like terms, we consolidate the equation into a more concise form, making it easier to isolate the variable. In our case, combining -5 and +6 to get 1 not only simplifies the equation but also clarifies the next steps needed to solve for x. This process of simplification ensures that we are working with the most straightforward representation of the equation, which minimizes the chances of errors and streamlines the solution process.
Moreover, simplification allows us to see the structure of the equation more clearly. We can easily identify the terms involving the variable and the constant terms, which helps in strategizing the next steps. For instance, after simplifying our equation to -42x + 1 = 10, it becomes evident that we need to isolate the term -42x before we can solve for x. This clarity is invaluable in maintaining focus and efficiency. Therefore, simplification is not just about reducing the number of terms but also about enhancing our understanding of the equation's components and their relationships, which ultimately leads to a more effective problem-solving approach.
Step 2: Isolate the Variable Term
Next, we need to isolate the variable term (-42x) on one side of the equation. To do this, we subtract 1 from both sides of the equation:
-42x + 1 - 1 = 10 - 1
-42x = 9
Isolating the variable term is a critical step in solving equations because it brings us closer to determining the value of the variable. This process involves performing inverse operations to eliminate any terms that are added to or subtracted from the variable term. By isolating -42x on the left side of the equation, we are effectively setting the stage for the final step, which involves dividing by the coefficient of x to solve for x itself.
The act of subtracting 1 from both sides of the equation maintains the equality, a fundamental principle in solving equations. This ensures that the solution we derive will satisfy the original equation. The operation is strategically chosen to counteract the +1 on the left side, thereby isolating the -42x term. This step-by-step approach is crucial for ensuring accuracy and minimizing the risk of algebraic errors. Furthermore, the clarity gained from isolating the variable term helps in visualizing the remaining steps required to reach the solution.
Step 3: Solve for x
To solve for x, we need to divide both sides of the equation by -42:
-42x / -42 = 9 / -42
x = -9/42
We can simplify the fraction -9/42 by dividing both the numerator and the denominator by their greatest common divisor, which is 3:
x = -3/14
Solving for x is the ultimate goal in this process, and it involves performing the necessary operation to get x by itself on one side of the equation. In our case, since -42 is multiplied by x, we divide both sides of the equation by -42. This division isolates x and gives us its value.
It is important to maintain precision when performing this step, as any error in division will lead to an incorrect solution. The negative sign also plays a crucial role here; dividing by -42 ensures that x is expressed as a positive or negative value as appropriate. Once we obtain the fraction -9/42, simplifying it to -3/14 is an essential step to express the solution in its simplest form. This simplification involves finding the greatest common divisor (GCD) of the numerator and denominator and dividing both by it. The simplified fraction not only makes the solution easier to understand but also reduces the likelihood of errors in subsequent calculations if the value of x is to be used in further steps or in related problems.
Step 4: Verify the Solution
To verify our solution, we substitute x = -3/14 back into the original equation:
-42(-3/14) - 5 + 6 = 10
First, multiply -42 by -3/14:
-42 * (-3/14) = 126/14 = 9
Now, substitute this back into the equation:
9 - 5 + 6 = 10
4 + 6 = 10
10 = 10
Since the equation holds true, our solution x = -3/14 is correct.
Verifying the solution is a critical step in the problem-solving process because it ensures that the value obtained for the variable satisfies the original equation. This step acts as a safeguard against errors made during the algebraic manipulations. By substituting the calculated value of x back into the original equation, we are essentially checking whether the left-hand side of the equation is equal to the right-hand side. If they are equal, the solution is correct; if not, there has been an error in the process, and we need to revisit the steps.
In our case, substituting x = -3/14 into the equation -42x - 5 + 6 = 10 involves careful arithmetic. We first multiply -42 by -3/14, which results in 9. Then, we substitute this value back into the equation, giving us 9 - 5 + 6. Simplifying this expression leads to 10, which matches the right-hand side of the equation. This confirms that our solution x = -3/14 is indeed correct. The verification step not only provides assurance that the problem has been solved correctly but also enhances understanding and confidence in the problem-solving approach.
Conclusion
In conclusion, solving the equation -42x - 5 + 6 = 10 involves simplifying the equation, isolating the variable term, solving for x, and verifying the solution. By following these steps, we found that x = -3/14 is the correct solution. Understanding these fundamental steps is essential for tackling more complex algebraic problems.
This methodical approach to solving linear equations provides a solid foundation for more advanced algebraic concepts. The ability to break down complex problems into manageable steps is a crucial skill in mathematics. Through practice and a clear understanding of these principles, one can confidently solve a wide range of algebraic equations. Remember, the key to mastering algebra is consistent practice and a thorough understanding of the underlying concepts.