Marlenas Equation Solving Journey A Step-by-Step Justification

In this article, we will delve into the step-by-step solution of the equation $2x + 5 = -10 - x$, as solved by Marlena. Our focus will be on providing a detailed justification for each step, ensuring a clear understanding of the underlying mathematical principles. This analysis is crucial for anyone looking to solidify their algebra skills or understand the logical progression involved in solving linear equations. We will break down each step, explaining the operations performed and the properties of equality that justify them. Understanding these justifications is key to mastering algebraic manipulations and solving a wide range of mathematical problems.

Step 1 Initial Equation $2x + 5 = -10 - x$

The initial equation presented is $2x + 5 = -10 - x$. This is our starting point, the foundation upon which we will build our solution. It's crucial to recognize this as a linear equation in one variable, 'x'. The goal is to isolate 'x' on one side of the equation to determine its value. Before we proceed, it's important to understand the structure of the equation. We have terms involving 'x' on both sides, as well as constant terms. Our objective is to manipulate the equation using valid algebraic operations to bring all 'x' terms to one side and all constant terms to the other. This process relies heavily on the properties of equality, which allow us to perform the same operation on both sides of the equation without changing its solution. By carefully applying these properties, we can systematically simplify the equation and ultimately solve for 'x'. The initial equation sets the stage for a series of steps, each designed to bring us closer to the solution. Understanding this initial setup is essential for following the subsequent steps and appreciating the logical flow of the solution process. Recognizing the type of equation and the variable we are solving for is the first critical step in any algebraic problem.

Step 2 Adding 'x' to Both Sides $3x + 5 = -10$

Marlena's second step involves adding 'x' to both sides of the equation. This is a crucial step in consolidating the 'x' terms on one side. The equation from Step 1 is $2x + 5 = -10 - x$. To eliminate the '-x' term on the right side, we apply the addition property of equality. This property states that adding the same quantity to both sides of an equation maintains the equality. Therefore, we add 'x' to both sides: $(2x + 5) + x = (-10 - x) + x$. On the left side, we combine like terms: $2x + x$ becomes $3x$. So the left side simplifies to $3x + 5$. On the right side, '-x' and '+x' cancel each other out, leaving just -10. This simplifies the right side to -10. Thus, the equation becomes $3x + 5 = -10$. This step is justified by the addition property of equality and the combination of like terms. It's a fundamental algebraic manipulation technique used to simplify equations and move closer to isolating the variable. By adding 'x' to both sides, Marlena has successfully eliminated the 'x' term from the right side, making the equation easier to solve. This demonstrates a clear understanding of algebraic principles and their application in equation solving.

Step 3 Subtracting 5 from Both Sides $3x = -15$

In the third step, Marlena subtracts 5 from both sides of the equation. This step aims to isolate the term with 'x' on one side. The equation from Step 2 is $3x + 5 = -10$. To eliminate the '+5' on the left side, we apply the subtraction property of equality. This property states that subtracting the same quantity from both sides of an equation maintains the equality. Therefore, we subtract 5 from both sides: $(3x + 5) - 5 = -10 - 5$. On the left side, '+5' and '-5' cancel each other out, leaving just $3x$. So the left side simplifies to $3x$. On the right side, we perform the subtraction: $-10 - 5$ equals $-15$. This simplifies the right side to $-15$. Thus, the equation becomes $3x = -15$. This step is justified by the subtraction property of equality. It's another fundamental algebraic manipulation technique used to further simplify equations and isolate the variable term. By subtracting 5 from both sides, Marlena has successfully isolated the term with 'x', bringing us closer to the final solution. This demonstrates a consistent application of algebraic principles in the equation-solving process. Isolating the variable term is a key strategy in solving equations, and this step effectively achieves that goal.

Step 4 Dividing Both Sides by 3 $x = -5$

The final step in Marlena's solution is dividing both sides of the equation by 3. This step isolates 'x' and provides the solution to the equation. The equation from Step 3 is $3x = -15$. To isolate 'x', we apply the division property of equality. This property states that dividing both sides of an equation by the same non-zero quantity maintains the equality. Therefore, we divide both sides by 3: $(3x) / 3 = (-15) / 3$. On the left side, 3 divided by 3 equals 1, leaving just $x$. So the left side simplifies to $x$. On the right side, we perform the division: $-15$ divided by 3 equals $-5$. This simplifies the right side to $-5$. Thus, the equation becomes $x = -5$. This step is justified by the division property of equality. It's the final algebraic manipulation needed to solve for 'x'. By dividing both sides by 3, Marlena has successfully isolated 'x' and determined its value. The solution to the equation $2x + 5 = -10 - x$ is $x = -5$. This demonstrates a complete and accurate application of algebraic principles in solving a linear equation. The final step confirms the value of 'x' that satisfies the original equation.

Conclusion Marlena's Solution Justified

In conclusion, Marlena's step-by-step solution to the equation $2x + 5 = -10 - x$ is logically sound and mathematically accurate. Each step is justified by fundamental properties of equality: the addition property, the subtraction property, and the division property. By adding 'x' to both sides, subtracting 5 from both sides, and finally dividing both sides by 3, Marlena successfully isolated 'x' and found the solution $x = -5$. This process demonstrates a clear understanding of algebraic principles and their application in solving linear equations. The ability to justify each step in an equation-solving process is crucial for building confidence and proficiency in algebra. Marlena's solution serves as a model for how to approach and solve linear equations systematically and accurately. Understanding these steps and their justifications is essential for anyone studying algebra or mathematics in general. The solution not only provides the answer but also illustrates the underlying mathematical reasoning, which is key to mastering the subject.