Justification For Step 3 Solving (23/10)h - (15/2) = 0

In the realm of mathematics, solving equations is a fundamental skill. It involves a systematic process of isolating the unknown variable to determine its value. This article delves into the solution of a specific equation, meticulously examining each step and providing a clear justification for the algebraic manipulations involved. Our focus will be on the equation presented, specifically dissecting Step 3 and elucidating the mathematical principle that underpins it. This exploration will not only enhance your understanding of equation-solving but also reinforce the importance of logical reasoning in mathematics. Let's embark on this journey of mathematical discovery, unraveling the steps involved and solidifying your grasp of algebraic problem-solving.

Understanding the Equation and the Steps

The equation we aim to solve is:

$\frac{23}{10} h - \frac{15}{2} = 0$

This equation involves a single variable, h, which we need to isolate to find its value. The provided solution outlines the steps taken to achieve this:

Step 1: $\frac{23}{10} h - \frac{15}{2} = 0$

This is the initial equation, setting the stage for our algebraic journey.

Step 2: $\frac{23}{10} h = \frac{15}{2}$

Here, we observe the addition of $\frac{15}{2}$ to both sides of the equation. This manipulation aims to isolate the term containing h on one side.

Step 3: $h = \frac{75}{23}$

This is the crucial step we will dissect and justify. It presents the final solution for h, but the method used to arrive at this value requires careful examination. Our primary focus will be to elucidate the algebraic operation that transforms Step 2 into Step 3, ensuring a comprehensive understanding of the solution process.

The Core Question: Justifying Step 3

The central question we address is: What is the justification for Step 3 in the solution process? To answer this, we must delve into the mathematical operation that transforms the equation from Step 2 to Step 3. Step 2 presents us with the equation $\frac{23}{10} h = \frac{15}{2}$. To isolate h, we need to eliminate the coefficient $\frac{23}{10}$ that is multiplying it. The key to achieving this lies in the concept of the multiplicative inverse.

The multiplicative inverse of a number is the value that, when multiplied by the original number, results in 1. For a fraction, the multiplicative inverse is simply the reciprocal – the fraction flipped. Therefore, the multiplicative inverse of $\frac{23}{10}$ is $\frac{10}{23}$.

To isolate h, we multiply both sides of the equation in Step 2 by $\frac{10}{23}$. This maintains the equality of the equation while effectively canceling out the coefficient of h on the left side. Let's break down this process:

• Start with Step 2: $\frac{23}{10} h = \frac{15}{2}$
• Multiply both sides by $\frac{10}{23}$: $\frac{10}{23} * \frac{23}{10} h = \frac{10}{23} * \frac{15}{2}$
• Simplify the left side: $1 * h = \frac{10}{23} * \frac{15}{2}$
• Simplify the left side: $h = \frac{10}{23} * \frac{15}{2}$

Now, we focus on simplifying the right side of the equation. We multiply the fractions:

$h = \frac{10 * 15}{23 * 2}$

$h = \frac{150}{46}$

Finally, we simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2:

$h = \frac{150 ÷ 2}{46 ÷ 2}$

$h = \frac{75}{23}$

Thus, we arrive at the solution presented in Step 3: $h = \frac{75}{23}$.

The Justification: Multiplicative Inverse

Therefore, the justification for Step 3 is the application of the multiplicative inverse property of multiplication. We multiplied both sides of the equation by the reciprocal of the coefficient of h to isolate the variable. This is a fundamental algebraic technique used to solve equations of this form. The use of the multiplicative inverse ensures that the equation remains balanced while allowing us to directly solve for the unknown variable. Understanding this principle is crucial for mastering equation-solving in algebra.

In summary, Step 3 is justified by multiplying both sides of the equation by the multiplicative inverse of $\frac{23}{10}$, which is $\frac{10}{23}$. This isolates h and leads to the solution $h = \frac{75}{23}$.

Why This Method Works: Maintaining Equality

The underlying principle that allows us to manipulate equations in this way is the concept of maintaining equality. An equation is a statement that two expressions are equal. To solve for a variable, we perform operations on both sides of the equation to isolate the variable while preserving the equality. Multiplying both sides by the same non-zero value is one such operation.

Think of an equation as a balanced scale. Both sides of the scale have the same weight. If we add or subtract the same weight from both sides, the scale remains balanced. Similarly, if we multiply or divide both sides by the same non-zero number, the scale remains balanced. This is the essence of algebraic manipulation.

In the context of our equation, multiplying both sides by $\frac{10}{23}$ is like multiplying the weight on both sides of the scale by the same factor. The balance, or equality, is maintained, but the form of the equation changes, bringing us closer to the solution. This concept of maintaining equality is a cornerstone of algebra and is crucial for understanding why we can perform certain operations on equations.

Alternative Approaches and Considerations

While multiplying by the multiplicative inverse is the most direct way to solve for h in this case, there are alternative approaches we could consider, although they might be less efficient.

One alternative would be to first eliminate the fractions by multiplying both sides of the equation in Step 1 by the least common multiple (LCM) of the denominators, which is 10. This would give us:

$10 * (\frac{23}{10} h - \frac{15}{2}) = 10 * 0$

$23h - 75 = 0$

Then, we would add 75 to both sides:

$23h = 75$

Finally, we would divide both sides by 23:

$h = \frac{75}{23}$

This approach arrives at the same solution but involves slightly more steps. It highlights the flexibility in algebraic manipulation and the importance of choosing the most efficient method.

Another consideration is the potential for errors. When working with fractions, it's crucial to be careful with arithmetic. Mistakes in multiplication or simplification can lead to an incorrect solution. Therefore, it's always a good practice to double-check your work, especially when dealing with fractions.

Furthermore, understanding the underlying principles, such as the multiplicative inverse property, helps in avoiding common errors. When you grasp the why behind the steps, you're less likely to make mistakes and more likely to solve equations accurately and efficiently.

The Significance of Step-by-Step Justification in Mathematics

In mathematics, providing a step-by-step justification for a solution is as important as arriving at the correct answer. It demonstrates a clear understanding of the underlying principles and the logical progression of the solution. Justification is not merely about showing how you arrived at the answer but also why each step is valid.

When we justify each step, we build a strong foundation of understanding. This allows us to tackle more complex problems with confidence and to apply these principles in different contexts. It also helps in identifying potential errors and correcting them. A well-justified solution is not only correct but also convincing and insightful.

In educational settings, teachers often emphasize the importance of showing your work and providing justifications. This is because the process of problem-solving is just as valuable as the solution itself. Justification encourages critical thinking, logical reasoning, and clear communication – skills that are essential not only in mathematics but also in many other areas of life.

Conclusion: Mastering the Art of Equation Solving

In conclusion, the justification for Step 3 in solving the equation $\frac{23}{10} h - \frac{15}{2} = 0$ lies in the application of the multiplicative inverse property. By multiplying both sides of the equation by $\frac{10}{23}$, we effectively isolated h and arrived at the solution $h = \frac{75}{23}$. This process highlights the importance of understanding fundamental algebraic principles and their role in equation-solving.

Mastering the art of equation-solving involves not only knowing the steps but also understanding the why behind them. By providing step-by-step justifications, we solidify our understanding, enhance our problem-solving skills, and build a strong foundation for further mathematical exploration. Remember, mathematics is not just about finding the right answer; it's about the journey of logical reasoning and the clear communication of mathematical ideas.

This detailed exploration of Step 3 serves as a testament to the power of logical reasoning in mathematics. By understanding the multiplicative inverse property and its application, we can confidently solve equations and tackle more complex mathematical challenges. The key is to break down each step, justify its validity, and build a solid understanding of the underlying principles. With practice and perseverance, anyone can master the art of equation-solving and unlock the beauty and power of mathematics.