Solving Equations Step-by-Step Examining Properties Of Equality

This article provides a detailed walkthrough of solving the equation $1.2x - 7.2x + 5 = -7$, with a strong emphasis on identifying the property of equality or reasoning applied at each step. Understanding these properties is crucial for mastering algebraic manipulations and solving more complex equations. Let's dive into the solution and dissect each step.

Step 1 Given Equation $1.2x - 7.2x + 5 = -7$

The first step in any equation-solving process is to state the given equation. In this case, we are presented with the equation $1.2x - 7.2x + 5 = -7$. This equation contains a variable, $x$, and our goal is to isolate $x$ on one side of the equation to find its value. This initial step simply acknowledges the starting point of our problem. No properties of equality are applied here; we are simply stating the given information.

Understanding the importance of the given equation is paramount in mathematics. It serves as the foundation upon which the entire solution is built. Every subsequent step must logically follow from this initial statement. Mistakes in transcribing or interpreting the given equation can lead to incorrect solutions. Therefore, it is always a good practice to double-check the given equation before proceeding with any algebraic manipulations. As we move forward, we'll see how the properties of equality allow us to manipulate this equation while maintaining its balance and ultimately revealing the value of $x$. This step sets the stage for the application of various algebraic principles and properties that will lead us to the solution. We will carefully examine each subsequent step to understand how these principles are employed. The given equation is not just a starting point; it's the cornerstone of the entire solution process.

Step 2 Combining Like Terms -6x + 5 = -7

In the second step, we simplify the left side of the equation by combining like terms. We have two terms involving $x$: $1.2x$ and $-7.2x$. Combining these gives us $(1.2 - 7.2)x = -6x$. The equation now becomes $-6x + 5 = -7$. This step utilizes the distributive property in reverse, where we factor out the common variable $x$. Combining like terms is a fundamental algebraic simplification technique. It allows us to reduce the complexity of the equation, making it easier to manipulate in subsequent steps. The distributive property, in its general form, states that $a(b + c) = ab + ac$. In our case, we are essentially reversing this process to combine the coefficients of the $x$ terms. This step is crucial because it streamlines the equation, bringing us closer to isolating the variable $x$. Furthermore, by combining like terms, we are making the equation more visually manageable, which reduces the chance of errors in later steps. This highlights the importance of simplifying equations as much as possible before attempting to isolate the variable.

Step 3 Applying the Subtraction Property of Equality -6x + 5 - 5 = -7 - 5

Here, we apply the subtraction property of equality. This property states that if we subtract the same value from both sides of an equation, the equation remains balanced. In this case, we subtract 5 from both sides of the equation to isolate the term with $x$. This gives us $-6x + 5 - 5 = -7 - 5$, which simplifies to $-6x = -12$. The subtraction property is a cornerstone of equation solving. It ensures that we maintain the equality while manipulating the equation to isolate the variable. Without this property, we wouldn't be able to perform operations on both sides of the equation without changing its fundamental nature. This step is particularly important because it removes the constant term from the left side, bringing us closer to isolating $x$. By subtracting 5 from both sides, we are effectively undoing the addition of 5 on the left side. This illustrates a key principle in equation solving: using inverse operations to isolate the variable. The subtraction property of equality, along with other properties of equality, forms the backbone of algebraic manipulation.

Step 4 Simplifying the Equation -6x = -12

This step is a direct result of the previous step, where we applied the subtraction property of equality. After subtracting 5 from both sides of the equation $-6x + 5 - 5 = -7 - 5$, we simplify it to $-6x = -12$. This step is a straightforward simplification, but it's crucial for clarity and accuracy. It sets the stage for the final step, where we will isolate $x$ by dividing both sides of the equation by -6. This simplified form of the equation clearly shows the relationship between -6 and x, making it easier to apply the next property of equality. While it might seem like a small step, this simplification helps to prevent errors and makes the solution process more transparent. Simplifying equations after each operation is a good practice to maintain clarity and reduce the complexity of the problem.

Step 5 Applying the Division Property of Equality -6x / -6 = -12 / -6

To isolate $x$, we now apply the division property of equality. This property states that if we divide both sides of an equation by the same non-zero value, the equation remains balanced. Here, we divide both sides of the equation $-6x = -12$ by -6. This gives us rac{-6x}{-6} = rac{-12}{-6}, which simplifies to $x = 2$. The division property of equality is another fundamental tool in equation solving. It allows us to undo multiplication, which is exactly what we need to do in this case. By dividing both sides by -6, we are effectively isolating $x$ on the left side. This step is the culmination of all the previous steps. It's where we finally isolate the variable and find its value. Without the division property, we wouldn't be able to solve for $x$ in this equation. This property, like the subtraction property, underscores the importance of performing the same operation on both sides of the equation to maintain balance.

Step 6 Solution x = 2

The final step is to state the solution. After dividing both sides of the equation by -6, we arrive at $x = 2$. This is the value of $x$ that satisfies the original equation. It's important to verify this solution by substituting it back into the original equation to ensure it holds true. Substituting $x = 2$ into $1.2x - 7.2x + 5 = -7$, we get $1.2(2) - 7.2(2) + 5 = 2.4 - 14.4 + 5 = -12 + 5 = -7$, which confirms our solution. This verification step is crucial for ensuring the accuracy of our solution. It's a safeguard against potential errors made during the algebraic manipulations. In conclusion, by systematically applying the properties of equality, we have successfully solved the equation and found that $x = 2$. This process highlights the importance of understanding and applying these properties correctly to solve algebraic equations.

Conclusion

Solving the equation $1.2x - 7.2x + 5 = -7$ involves a series of steps, each relying on fundamental properties of equality. We began by simplifying the equation by combining like terms, then used the subtraction property of equality to isolate the term with $x$. Finally, we applied the division property of equality to solve for $x$. Understanding these properties is essential for successfully manipulating equations and finding solutions. Each step in the process is a logical consequence of the previous one, demonstrating the power and elegance of algebraic reasoning. By carefully applying these principles, we can confidently solve a wide range of algebraic equations. This step-by-step examination not only provides the solution but also reinforces the importance of understanding the underlying mathematical principles involved in equation solving. Mastering these concepts is key to success in algebra and beyond.

What properties of equality are used in each step of solving the equation 1.2x - 7.2x + 5 = -7?

Solving Equations Step-by-Step Examining Properties of Equality