Justification For Step 3 In Solving Linear Equations An In Depth Explanation

Jul 9, 2025 by ADMIN 77 views

Solving Linear Equations A Detailed Explanation of Step 3 in 0.8a - 0.1a = a - 2.5

Introduction

In the realm of mathematics, particularly in algebra, solving linear equations is a fundamental skill. Understanding each step involved in the solution process is crucial for mastering this skill. This article delves into the step-by-step solution of the linear equation 0.8a - 0.1a = a - 2.5, with a specific focus on justifying step 3. We will break down the equation, explain the logic behind each operation, and highlight the mathematical properties that validate each step. This detailed exploration aims to provide a comprehensive understanding of solving linear equations and the principles that govern them.

Problem Statement

We are given the linear equation:

0.8a - 0.1a = a - 2.5

The solution process is outlined in three steps:

Step 1: 0.7a = a - 2.5
Step 2: -0.3a = -2.5
Step 3: a = 8.3

Our primary goal is to thoroughly justify step 3, but to fully appreciate its significance, we will first revisit steps 1 and 2, ensuring a clear understanding of the entire solution process.

Step-by-Step Solution and Justification

Step 1: Simplifying the Left-Hand Side

The initial equation is 0.8a - 0.1a = a - 2.5. In step 1, we simplify the left-hand side (LHS) of the equation. Both terms, 0.8a and 0.1a, contain the variable a, which means they are like terms and can be combined. The operation involves subtracting the coefficients: 0.8 - 0.1 = 0.7. Therefore, the left-hand side simplifies to 0.7a. This simplification is justified by the distributive property in reverse, which allows us to combine like terms. The distributive property states that for any numbers x, y, and z, x(y + z) = xy + xz. In our case, we are applying this in reverse: 0.8a - 0.1a = (0.8 - 0.1)a = 0.7a. This step is straightforward and crucial for making the equation more manageable.

Step 2: Isolating the Variable Term

After step 1, our equation becomes 0.7a = a - 2.5. To continue solving for a, we need to isolate the terms containing a on one side of the equation. In step 2, we subtract a from both sides of the equation. This operation is justified by the subtraction property of equality, which states that if x = y, then x - z = y - z for any numbers x, y, and z. Applying this property, we subtract a from both sides:

0.  7a - a = a - 2.5 - a

On the left-hand side, 0.7a - a can be rewritten as 0.7a - 1a. Subtracting the coefficients, 0.7 - 1 = -0.3, so the left-hand side simplifies to -0.3a. On the right-hand side, a - a cancels out, leaving -2.5. Thus, the equation becomes -0.3a = -2.5. This step is vital because it moves all terms containing the variable to one side, bringing us closer to the solution.

Step 3: Solving for 'a'

Step 3 is the culmination of the previous steps, where we finally solve for the variable a. The equation at the beginning of this step is -0.3a = -2.5. To isolate a, we need to undo the multiplication by -0.3. This is achieved by dividing both sides of the equation by -0.3. This operation is justified by the division property of equality, which states that if x = y, then x / z = y / z for any numbers x, y, and non-zero z. Applying this property, we divide both sides by -0.3:

(-0.3a) / (-0.3) = (-2.5) / (-0.3)

On the left-hand side, -0.3a divided by -0.3 simplifies to a, as any non-zero number divided by itself is 1. On the right-hand side, we perform the division: -2.5 / -0.3. Dividing a negative number by a negative number results in a positive number. To perform this division, we can rewrite the expression as a fraction: 2.5 / 0.3. To eliminate the decimals, we can multiply both the numerator and the denominator by 10:

(2.  5 * 10) / (0.3 * 10) = 25 / 3

Now, we divide 25 by 3, which gives us a mixed number or a decimal. As a mixed number, it is 8 1/3. As a decimal, it is approximately 8.333..., which can be written as 8.3 with a bar over the 3 to indicate that the 3 repeats infinitely. Therefore, the solution for a is 8.3.

In summary, step 3 is justified by the division property of equality. This property allows us to divide both sides of an equation by the same non-zero number without changing the solution. This step is the final step in isolating the variable and finding its value.

Conclusion

In this article, we have thoroughly examined the solution process for the linear equation 0.8a - 0.1a = a - 2.5, with a particular emphasis on justifying step 3. We saw that step 3, where we divided both sides of the equation by -0.3, is justified by the division property of equality. This property, along with the subtraction property of equality and the distributive property, forms the backbone of solving linear equations. By understanding these properties and the steps involved, one can confidently tackle a wide range of algebraic problems. Mastering these concepts is essential for success in mathematics and related fields.

Through this detailed explanation, we hope to have provided a clear understanding of not just the mechanics of solving the equation, but also the underlying mathematical principles that make it valid. This approach to learning mathematics, focusing on both the how and the why, is key to developing a deep and lasting understanding.