Paula's Equation Solving: Identify Division Property Application

Let's analyze Paula's equation-solving steps to pinpoint where the division property of equality was applied. Understanding this property is crucial for mastering algebraic manipulations. In this detailed explanation, we will break down each step, highlight the underlying principles, and clarify the exact moment Paula used the division property.

The Importance of the Division Property of Equality

The division property of equality is a fundamental concept in algebra. It states that if you divide both sides of an equation by the same non-zero number, the equation remains balanced. In simpler terms, the solutions to the equation remain unchanged. This property is essential for isolating the variable and finding its value. The power of this property lies in its ability to simplify equations while preserving their integrity. Applying it correctly is key to solving algebraic problems accurately.

Step-by-Step Breakdown of Paula's Solution

To clearly identify when the division property was used, let's meticulously examine each step of Paula's solution. By understanding the transitions between steps, we can better appreciate the role of the division property in the overall process.

Step 1: Initial Equation $-4(x+8)-2x = 25$

Paula starts with the equation $-4(x+8)-2x = 25$. This equation presents a mix of terms and parentheses, making it a good example of an algebraic problem that requires simplification. The initial task is to prepare the equation for solving by dealing with the parentheses and combining like terms.

Step 2: Applying the Distributive Property $-4x - 32 - 2x = 25$

In Step 2, Paula applies the distributive property. She multiplies -4 by both x and 8 inside the parentheses: $-4 * x = -4x$ and $-4 * 8 = -32$. The equation becomes $-4x - 32 - 2x = 25$. This step is a crucial simplification, as it removes the parentheses and allows for combining like terms.

Step 3: Combining Like Terms $-6x - 32 = 25$

Here, Paula combines the like terms, which are the terms containing x. She adds $-4x$ and $-2x$ to get $-6x$. The equation is now simplified to $-6x - 32 = 25$. This step makes the equation more manageable and closer to a form where the variable can be isolated.

Step 4: Isolating the Variable Term $-6x = 57$

To isolate the term with x, Paula adds 32 to both sides of the equation. This cancels out the -32 on the left side, leaving $-6x = 25 + 32$, which simplifies to $-6x = 57$. This step is an application of the addition property of equality, a counterpart to the division property, where adding the same value to both sides maintains the equation's balance. Isolating the variable term is a critical move toward finding the value of x.

Step 5: Applying the Division Property x = -9rac{1}{2}

In Step 5, Paula applies the division property of equality. She divides both sides of the equation by -6 to isolate x: $\frac{-6x}{-6} = \frac{57}{-6}$. This simplifies to $x = -\frac{57}{6}$, which further reduces to $x = -9\frac{1}{2}$. This is the final step where the division property is used to solve for x. The division property ensures that the solution obtained is correct by maintaining the balance of the equation.

Identifying the Division Property: Steps 4 and 5

The division property of equality is employed specifically between Step 4 ($-6x = 57$) and Step 5 ($x = -9\frac{1}{2}$). This is the crucial transition where both sides of the equation are divided by -6 to isolate x.

Why This Step is the Division Property

To reiterate, the division property is the act of dividing both sides of an equation by the same non-zero value. In Paula's solution, this occurs when she divides both sides of $-6x = 57$ by -6. This action perfectly fits the definition of the division property and is the key to solving for x.

Common Mistakes and How to Avoid Them

When using the division property, a common mistake is forgetting to divide every term on both sides of the equation. Another error is dividing by zero, which is undefined and invalidates the solution. Always ensure you are dividing by a non-zero constant and that the operation is applied uniformly across the equation. Furthermore, it's vital to correctly handle negative signs to avoid sign errors in the final answer.

Reinforcing the Concept with Examples

To solidify understanding, let's consider a few more examples where the division property is essential:

1. Solve $5x = 20$: Divide both sides by 5 to get $x = 4$.
2. Solve $-3y = 18$: Divide both sides by -3 to get $y = -6$.
3. Solve $2z = -24$: Divide both sides by 2 to get $z = -12$.

These examples demonstrate how the division property is applied in straightforward linear equations.

Advanced Applications of the Division Property

The division property is not limited to simple equations; it is also used in more complex scenarios. For instance, in multi-step equations, it often follows simplification and combining like terms. It is also a key step in solving systems of equations and in various areas of calculus and advanced algebra. The ability to correctly apply the division property is a cornerstone of mathematical problem-solving.

Conclusion: Mastering the Division Property

In summary, Paula used the division property of equality between Step 4 and Step 5 of her solution. This property is a fundamental tool in algebra, allowing us to isolate variables and solve equations accurately. By understanding and applying this property correctly, we can tackle a wide range of algebraic problems with confidence. Remembering the basic principles and avoiding common mistakes will pave the way for success in algebra and beyond. The division property is more than just a step in solving equations; it's a key concept that underpins much of algebraic manipulation.

Repair Input Keyword

Between which steps did Paula apply the division property of equality to solve the equation?

SEO Title

Paula's Equation Solving Identify Division Property Application