Justification For Step 2 Solving Algebraic Equation

by ADMIN 52 views

In the realm of mathematics, solving equations is a fundamental skill that underpins various disciplines. A meticulous, step-by-step approach is crucial for achieving accurate results. When confronted with an equation, each step must be logically sound and justified by mathematical principles. In this article, we delve into the process of solving an algebraic equation, focusing on the justification behind a specific step. Our exploration centers around the equation:

12r+12=−27r+67−5\frac{1}{2} r + \frac{1}{2} = -\frac{2}{7} r + \frac{6}{7} - 5

We will analyze the provided steps, paying particular attention to Step 2, to unravel the mathematical reasoning behind it. Our aim is to provide a comprehensive understanding of the mathematical operations and principles employed in the solution process.

To fully grasp the justification for Step 2, it's essential to examine the initial equation and the transformations applied to it. Let's break down the steps:

Step 1: Simplifying the Right-Hand Side

The initial equation is:

12r+12=−27r+67−5\frac{1}{2} r + \frac{1}{2} = -\frac{2}{7} r + \frac{6}{7} - 5

Step 1 involves simplifying the right-hand side of the equation. Specifically, the constant terms 67\frac{6}{7} and −5-5 are combined. To do this, we need to express −5-5 as a fraction with a denominator of 7. Thus, −5-5 becomes −357-\frac{35}{7}. Now, we can combine the fractions:

67−357=6−357=−297\frac{6}{7} - \frac{35}{7} = \frac{6 - 35}{7} = -\frac{29}{7}

Therefore, Step 1 transforms the equation into:

12r+12=−27r−297\frac{1}{2} r + \frac{1}{2} = -\frac{2}{7} r - \frac{29}{7}

This step is justified by the basic arithmetic operation of combining like terms. It streamlines the equation, making it easier to work with in subsequent steps. The key here is to maintain the equality by performing the same operations on one side of the equation as on the other.

Step 2: What is the Justification for Step 2 in the solution process?

Now, let's address the core question: What is the justification for Step 2? Unfortunately, the provided text only gives us Step 1 and the question about Step 2. To answer this question comprehensively, we need to infer what Step 2 might be and then provide the justification. A logical next step in solving this equation would be to eliminate the fractions to simplify the equation further. This can be achieved by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, which are 2 and 7. The LCM of 2 and 7 is 14.

So, let's assume Step 2 involves multiplying both sides of the equation by 14:

14(12r+12)=14(−27r−297)14 \left( \frac{1}{2} r + \frac{1}{2} \right) = 14 \left( -\frac{2}{7} r - \frac{29}{7} \right)

Now, we distribute the 14 on both sides:

14⋅12r+14⋅12=14⋅(−27r)+14⋅(−297)14 \cdot \frac{1}{2} r + 14 \cdot \frac{1}{2} = 14 \cdot \left( -\frac{2}{7} r \right) + 14 \cdot \left( -\frac{29}{7} \right)

This simplifies to:

7r+7=−4r−587r + 7 = -4r - 58

The justification for this step lies in the Multiplication Property of Equality. This property states that if you multiply both sides of an equation by the same non-zero number, the equation remains balanced. In other words, the solutions to the equation do not change. By multiplying by the LCM, we eliminate the fractions, making the equation easier to solve without altering its fundamental nature.

Eliminating fractions is a common and powerful technique in algebra. It transforms an equation with fractions into an equivalent equation without fractions, which is often simpler to manipulate and solve. This step is crucial for efficiently finding the value of the variable r.

Subsequent Steps (Hypothetical)

While the question focuses on Step 2, it's beneficial to outline the subsequent steps to complete the solution process. These steps would likely involve:

  1. Moving all terms with r to one side of the equation and constant terms to the other side. This is achieved by adding 4r4r to both sides:

    7r+4r+7=−4r+4r−587r + 4r + 7 = -4r + 4r - 58

    11r+7=−5811r + 7 = -58

    The justification here is the Addition Property of Equality, which states that adding the same quantity to both sides of an equation preserves equality.

  2. Subtracting 7 from both sides:

    11r+7−7=−58−711r + 7 - 7 = -58 - 7

    11r=−6511r = -65

    Again, the Addition Property of Equality (in this case, adding a negative number) justifies this step.

  3. Dividing both sides by 11 to isolate r:

    11r11=−6511\frac{11r}{11} = \frac{-65}{11}

    r=−6511r = -\frac{65}{11}

    This step is justified by the Division Property of Equality, which states that dividing both sides of an equation by the same non-zero number maintains the equality.

The emphasis on justifying each step in the solution process is not merely a matter of formality; it is fundamental to the integrity and accuracy of mathematical reasoning. Justification ensures that each transformation of the equation is valid and that the final solution is logically derived from the initial equation. This approach fosters a deeper understanding of mathematical principles and enhances problem-solving skills.

When solving equations, it's crucial to be aware of the properties of equality, such as the Addition, Subtraction, Multiplication, and Division Properties. These properties provide the foundation for manipulating equations while preserving their solutions. Understanding these justifications allows us to confidently navigate complex mathematical problems.

In the process of solving equations, several common mistakes can occur. One frequent error is failing to apply the same operation to both sides of the equation, thereby disrupting the balance and leading to an incorrect solution. For instance, if we were to multiply only the left side of the equation by 14 in Step 2, the equality would no longer hold.

Another common mistake is incorrectly applying the distributive property when multiplying a constant by an expression in parentheses. It is essential to ensure that the constant is multiplied by each term within the parentheses. A failure to do so will lead to errors in the solution.

To avoid these mistakes, it is vital to meticulously check each step and ensure that the operations performed are consistent with the properties of equality. Additionally, it is helpful to rewrite the equation after each step to visually confirm that the transformations are correct. Practice and attention to detail are key to mastering equation-solving skills.

In conclusion, the justification for Step 2, which we inferred to be multiplying both sides of the equation by the least common multiple of the denominators, is rooted in the Multiplication Property of Equality. This property allows us to eliminate fractions, simplifying the equation without altering its solutions. Understanding the justifications behind each step in the solution process is crucial for developing strong mathematical skills and avoiding errors.

By meticulously applying mathematical principles and properties, we can confidently solve equations and tackle more complex mathematical challenges. The ability to justify each step not only ensures accuracy but also fosters a deeper understanding of the underlying mathematical concepts. This approach is essential for success in mathematics and related fields.

  • Solving equations
  • Justification in mathematics
  • Multiplication Property of Equality
  • Algebraic equations
  • Least common multiple
  • Eliminating fractions
  • Mathematical problem solving
  • Properties of equality
  • Step-by-step solution
  • Equation solving techniques