Marlena's Equation Solving Journey Step-by-Step Justification

Marlena embarked on a mathematical journey to solve the equation $2x + 5 = -10 - x$. Her meticulous steps are outlined below, and we will delve into the justification behind each step, unraveling the logic and mathematical principles employed.

$2x + 5 = -10 - x$

1. $3x + 5 = -10$
2. $3x = -15$
3. $x = -5$

Let's dissect Marlena's work step by step, providing the mathematical rationale for each transformation. We'll explore the fundamental properties of equality that underpin her solution, ensuring a clear understanding of the process.

Step 1: Unveiling the Addition Property of Equality

In Step 1, Marlena transforms the equation from $2x + 5 = -10 - x$ to $3x + 5 = -10$. The key to understanding this step lies in the Addition Property of Equality. This property states that adding the same value to both sides of an equation maintains the equality. In simpler terms, if we have two things that are equal, adding the same amount to both will keep them equal.

Marlena strategically added '$x$' to both sides of the original equation. This maneuver is crucial because it aims to consolidate the 'x' terms on one side of the equation. By adding '$x$' to both $2x$ and -$x$, she effectively eliminates the 'x' term from the right side of the equation and combines it with the 'x' term on the left side.

Let's break it down:

• Original Equation: $2x + 5 = -10 - x$
• Adding '$x$' to both sides: $2x + 5 + x = -10 - x + x$
• Simplifying: $3x + 5 = -10$

As you can see, adding '$x$' to both sides allowed Marlena to combine the 'x' terms on the left, resulting in $3x$. On the right side, the '-$x$' and '+$x$' canceled each other out, leaving just -10. This single step brought her closer to isolating 'x', which is the ultimate goal when solving equations.

The Addition Property of Equality is a cornerstone of algebraic manipulation. It allows us to rearrange equations without altering their fundamental truth. By judiciously adding terms to both sides, we can simplify equations and bring them into a form that reveals the solution more readily. Marlena's application of this property in Step 1 demonstrates a clear understanding of this core principle.

Step 2: Isolating the Variable Through Subtraction

Step 2 marks a pivotal point in Marlena's solution. She transitions from $3x + 5 = -10$ to $3x = -15$. This transformation hinges on the Subtraction Property of Equality, a close cousin to the addition property we discussed earlier. The Subtraction Property of Equality dictates that subtracting the same value from both sides of an equation preserves the balance, maintaining the equality.

In this instance, Marlena's objective is to isolate the term containing 'x', which is $3x$. To achieve this, she needs to eliminate the constant term '+5' that is currently cluttering the left side of the equation. The strategic move here is to subtract '5' from both sides.

Let's dissect the process:

• Starting Equation: $3x + 5 = -10$
• Subtracting '5' from both sides: $3x + 5 - 5 = -10 - 5$
• Simplifying: $3x = -15$

As we observe, subtracting '5' from the left side effectively cancels out the '+5', leaving us with just $3x$. On the right side, subtracting '5' from '-10' results in '-15'. This seemingly simple step is crucial because it isolates the term with 'x', bringing us closer to the final solution.

The Subtraction Property of Equality is an indispensable tool in the equation-solving arsenal. It empowers us to strategically remove unwanted terms from one side of the equation, paving the way for isolating the variable of interest. Marlena's skillful application of this property in Step 2 showcases her understanding of how to manipulate equations while upholding their fundamental balance.

Step 3: The Final Act: Division Property of Equality

Step 3 represents the culmination of Marlena's equation-solving process. She elegantly moves from $3x = -15$ to the conclusive answer, $x = -5$. The guiding principle behind this final step is the Division Property of Equality. This property asserts that dividing both sides of an equation by the same non-zero value maintains the equality. It's a fundamental principle that allows us to scale down or scale up both sides of an equation without disrupting the balance.

Marlena's goal in this step is crystal clear: to isolate 'x' completely. Currently, 'x' is shackled to a coefficient of '3'. To liberate 'x', she needs to undo the multiplication by '3'. This is where the Division Property of Equality comes into play. By dividing both sides of the equation by '3', she effectively cancels out the coefficient and reveals the value of 'x'.

Let's examine the mechanics:

• Equation at the start of the step: $3x = -15$
• Dividing both sides by '3': $(3x) / 3 = -15 / 3$
• Simplifying: $x = -5$

As we see, dividing $3x$ by '3' neatly isolates 'x'. On the right side, dividing '-15' by '3' yields '-5'. Thus, Marlena arrives at the solution: $x = -5$. This value represents the point where the original equation holds true.

The Division Property of Equality is a powerful technique for unwinding multiplication and revealing the value of a variable. It's the final piece of the puzzle in many equation-solving scenarios. Marlena's deft application of this property in Step 3 underscores her mastery of algebraic manipulation and her ability to arrive at accurate solutions.

In conclusion, Marlena's journey to solve the equation $2x + 5 = -10 - x$ exemplifies the strategic use of fundamental algebraic principles. Each step is meticulously justified by the properties of equality, showcasing a clear understanding of how to manipulate equations while preserving their balance. From the Addition Property to the Subtraction Property and finally the Division Property, Marlena's work provides a valuable lesson in the art of equation solving.