Solving For X In The Equation -x + 5 = -5

When dealing with linear equations, our primary goal is to isolate the variable we're trying to solve for. In this case, we want to find the value of x that satisfies the equation -x + 5 = -5. This involves using algebraic manipulations to get x by itself on one side of the equation. Each step we take must maintain the equality, ensuring that both sides of the equation remain balanced. We achieve this by performing the same operation on both sides, whether it's adding, subtracting, multiplying, or dividing. The key is to systematically undo the operations that are applied to x, effectively peeling away the layers until x stands alone.

The process begins by identifying the terms that are associated with x. In the equation -x + 5 = -5, we see that x is being multiplied by -1 (implicitly) and then having 5 added to it. Our strategy is to reverse these operations in the opposite order. First, we'll address the addition of 5, and then we'll deal with the multiplication by -1. This step-by-step approach is crucial in solving linear equations accurately and efficiently. By following this method, we can confidently navigate through the equation and arrive at the correct solution for x.

The beauty of algebra lies in its ability to transform equations while preserving their underlying truth. Each manipulation we perform is a step towards clarity, revealing the value of the unknown variable. As we progress through the solution, we'll be employing fundamental algebraic principles, such as the addition and multiplication properties of equality. These properties are the bedrock of equation solving, allowing us to move terms around and simplify expressions without altering the equation's balance. With each step, we'll inch closer to isolating x and uncovering its value, demonstrating the power and elegance of algebraic techniques.

Step-by-Step Solution

1. Subtract 5 from both sides

To isolate the term containing x, we first need to eliminate the constant term on the left side of the equation. In the equation -x + 5 = -5, the constant term is 5. To remove it, we perform the inverse operation, which is subtraction. We subtract 5 from both sides of the equation, ensuring that the equality is maintained. This step is crucial in simplifying the equation and bringing us closer to isolating x. By subtracting 5 from both sides, we effectively cancel out the 5 on the left side, leaving us with just the term containing x. This process highlights the importance of performing the same operation on both sides of an equation to maintain its balance and arrive at the correct solution.

Mathematically, this looks like:

-x + 5 - 5 = -5 - 5

Simplifying this gives us:

-x = -10

2. Multiply both sides by -1

Now that we have -x = -10, we need to solve for x itself, not -x. The negative sign in front of x implies that x is being multiplied by -1. To isolate x, we perform the inverse operation, which is multiplication by -1. Multiplying both sides of the equation by -1 will change the sign of both sides, effectively removing the negative sign from x. This step is essential in obtaining the positive value of x, which is what we're ultimately trying to find. By multiplying both sides by -1, we ensure that the equality is maintained while simultaneously isolating x. This process demonstrates the power of algebraic manipulation in solving equations and revealing the value of the unknown variable.

This gives us:

(-1) * (-x) = (-1) * (-10)

Simplifying this results in:

x = 10

Final Answer

Therefore, the value of x that makes the equation -x + 5 = -5 true is 10. This solution is obtained by systematically isolating x through algebraic manipulations, ensuring that each step maintains the equality of the equation. By subtracting 5 from both sides and then multiplying both sides by -1, we effectively peel away the layers surrounding x, revealing its value. This process highlights the fundamental principles of equation solving and demonstrates the power of algebra in finding unknown quantities.

To verify our solution, we can substitute x = 10 back into the original equation and check if it holds true. If the left side of the equation equals the right side when x = 10, then our solution is correct. This step is crucial in ensuring the accuracy of our answer and provides a sense of confidence in our problem-solving abilities. Verification is a fundamental aspect of mathematical practice, allowing us to confirm our results and deepen our understanding of the concepts involved. In this case, substituting 10 for x in the original equation will confirm that our solution is indeed correct.

Substituting x = 10 into the original equation:

-x + 5 = -5

-10 + 5 = -5

-5 = -5

Since the left side equals the right side, our solution is correct. Therefore, the value of x that makes the equation true is 10.