Paula's Equation Solving Journey Unveiling The Division Property Of Equality

In this article, we'll dissect Paula's step-by-step solution to an equation, pinpointing exactly where she employed the division property of equality. Understanding this property is crucial for mastering algebraic manipulations and solving equations effectively. We'll break down each step, highlighting the key operations and reasoning behind them. So, let's embark on this mathematical journey and unravel the mystery of the division property!

Paula's Equation-Solving Journey

Paula embarked on a mission to solve the equation $-4(x+8)-2x=25$. Her journey, meticulously laid out in five steps, showcases the power of algebraic manipulation. Let's retrace her steps and analyze each transformation, paying close attention to the properties she employed along the way.

Step 1: The Initial Equation

Paula started with the equation:

-4(x+8)-2x=25

This is our starting point, the equation we aim to solve for the unknown variable, x. The equation presents a combination of terms involving parentheses, multiplication, and subtraction. To isolate x, Paula needs to carefully dismantle these operations, one step at a time.

Step 2: Distributing the Magic

The transition from Step 1 to Step 2 reveals a crucial move: the application of the distributive property. Paula multiplied the -4 across the terms inside the parentheses (x + 8). This means multiplying -4 by x and -4 by 8:

-4 * x = -4x
-4 * 8 = -32

This yields the equation:

-4x - 32 - 2x = 25

The distributive property is a cornerstone of algebra, allowing us to simplify expressions by multiplying a term across a sum or difference within parentheses. It's a fundamental technique for unwrapping expressions and making them easier to work with. Notice how the equation is becoming less cluttered, paving the way for further simplification.

Step 3: Combining Like Terms

In Step 3, Paula combined the like terms on the left side of the equation. Like terms are those that have the same variable raised to the same power. In this case, -4x and -2x are like terms. Combining them involves adding their coefficients:

-4x + (-2x) = -6x

This simplifies the equation to:

-6x - 32 = 25

Combining like terms is another essential simplification technique. It reduces the number of terms in the equation, making it more manageable and bringing us closer to isolating the variable x. The equation now presents a cleaner picture, with fewer moving parts.

Step 4: Isolating the Variable Term

Step 4 marks a significant move towards isolating x. Paula added 32 to both sides of the equation. This is based on the addition property of equality, which states that adding the same value to both sides of an equation maintains the equality. The goal here is to eliminate the -32 on the left side:

-6x - 32 + 32 = 25 + 32

This simplifies to:

-6x = 57

The addition property of equality is a powerful tool for manipulating equations. By strategically adding values to both sides, we can shift terms around and isolate the variable we're trying to solve for. The variable term, -6x, is now standing alone on the left side, ready for the final step.

Step 5: The Division Property Unveiled

Finally, in Step 5, Paula makes the crucial move that utilizes the division property of equality. To isolate x, she divides both sides of the equation by -6:

(-6x) / -6 = 57 / -6

This simplifies to:

x = -9.5

Or, expressed as a mixed number:

x = -9 \frac{1}{2}

The division property of equality is the counterpart to the multiplication property. It allows us to divide both sides of an equation by the same non-zero value without disturbing the balance. This step effectively undoes the multiplication by -6, leaving x isolated and revealing its value.

The Division Property of Equality: The Key to Isolation

The division property of equality is a fundamental principle in algebra. It asserts that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. This property is instrumental in isolating variables and solving for their values. It's one of the core tools in an algebraist's toolkit, enabling us to unravel equations and find solutions.

Let's delve deeper into the concept. Imagine an equation as a balanced scale. The left side represents one pan, and the right side represents the other. The equals sign (=) signifies that the scale is perfectly balanced. To maintain this balance, any operation performed on one side must be mirrored on the other. The division property exemplifies this principle perfectly. If you divide the weight on one pan by a certain number, you must divide the weight on the other pan by the same number to keep the scale balanced. This ensures that the equation remains true.

The division property is especially crucial when the variable we're trying to solve for is being multiplied by a coefficient. A coefficient is simply the number that's multiplying the variable (e.g., in the term 5x, 5 is the coefficient). To isolate the variable, we need to undo this multiplication. This is precisely where the division property comes into play. By dividing both sides of the equation by the coefficient, we effectively "cancel out" the multiplication and leave the variable by itself.

For instance, consider the equation 3x = 12. Our goal is to find the value of x. To do this, we need to isolate x on one side of the equation. Notice that x is being multiplied by 3. To undo this multiplication, we apply the division property of equality. We divide both sides of the equation by 3:

(3x) / 3 = 12 / 3

This simplifies to:

x = 4

Therefore, the solution to the equation 3x = 12 is x = 4. The division property allowed us to isolate x and determine its value.

The division property isn't just a standalone tool; it often works in tandem with other properties of equality, such as the addition, subtraction, and multiplication properties. Together, these properties form a powerful arsenal for solving a wide range of algebraic equations. Mastering the division property is therefore an essential step in building a solid foundation in algebra.

In summary, the division property of equality is a cornerstone of algebraic manipulation. It ensures that equations remain balanced while allowing us to isolate variables and solve for their values. By dividing both sides of an equation by the same non-zero number, we can effectively undo multiplication and pave the way for finding solutions. This property, along with its counterparts, is indispensable for anyone seeking to conquer the world of algebra.

Pinpointing the Division Property's Application

To definitively answer the question, Paula used the division property of equality between Step 4 and Step 5. In Step 4, the equation was -6x = 57. To isolate x, Paula divided both sides by -6, which is the essence of the division property, leading to the solution x = -9 1/2 in Step 5.

Conclusion: Mastering the Properties of Equality

Paula's equation-solving journey beautifully illustrates the power of algebraic properties, particularly the division property of equality. By understanding and applying these properties, we can systematically unravel equations and find solutions. This problem serves as a valuable reminder of the importance of each step in the algebraic process and how each property plays a crucial role in reaching the final answer. Mastering these properties is key to success in algebra and beyond.