Solving 0.025x + 3.75 = 1.4 A Step-by-Step Guide

Jul 14, 2025 by ADMIN 49 views

Solving the Equation 0.025x + 3.75 = 1.4: A Comprehensive Guide

In the realm of mathematics, equations serve as fundamental tools for problem-solving and understanding relationships between variables. This article delves into the process of solving a specific equation: 0.025x + 3.75 = 1.4. We will embark on a step-by-step journey, unraveling the equation's intricacies and arriving at a solution. This guide not only provides the answer but also equips you with the knowledge and skills to tackle similar mathematical challenges with confidence.

Understanding the Equation

Before we dive into the solution, let's take a moment to understand the equation's structure. The equation 0.025x + 3.75 = 1.4 is a linear equation, which means it involves a variable (in this case, 'x') raised to the power of 1. The equation states that the expression 0.025x + 3.75 is equal to 1.4. Our goal is to isolate the variable 'x' on one side of the equation to determine its value.

Key components of the equation:

Variable: 'x' represents the unknown value we aim to find.
Coefficient: 0.025 is the coefficient of 'x', indicating that 'x' is multiplied by this value.
Constant terms: 3.75 and 1.4 are constant terms, representing fixed numerical values.

The equation essentially poses a question: "What value of 'x' will make the expression 0.025x + 3.75 equal to 1.4?" To answer this question, we will employ algebraic techniques to manipulate the equation and isolate 'x'.

Step-by-Step Solution

Now, let's embark on the journey of solving the equation step by step. We will follow a systematic approach, ensuring that each step is logically sound and brings us closer to the solution.

Step 1: Isolate the term with 'x'

The first step is to isolate the term containing the variable 'x' (which is 0.025x) on one side of the equation. To achieve this, we need to eliminate the constant term (+3.75) from the left side. We can do this by subtracting 3.75 from both sides of the equation. This maintains the equation's balance, as we are performing the same operation on both sides.

Original equation:

0.  025x + 3.75 = 1.4

Subtract 3.75 from both sides:

0.  025x + 3.75 - 3.75 = 1.4 - 3.75

Simplify:

0.  025x = -2.35

Now, we have successfully isolated the term 0.025x on the left side of the equation. The equation now reads 0.025x = -2.35.

Step 2: Solve for 'x'

The next step is to isolate 'x' itself. Currently, 'x' is being multiplied by 0.025. To undo this multiplication, we need to divide both sides of the equation by 0.025. Again, this maintains the equation's balance.

Equation:

0.  025x = -2.35

Divide both sides by 0.025:

(0.025x) / 0.025 = -2.35 / 0.025

Simplify:

x = -94

Therefore, the solution to the equation 0.025x + 3.75 = 1.4 is x = -94.

Verification

To ensure the accuracy of our solution, it's always a good practice to verify it. We can do this by substituting the value we found for 'x' back into the original equation and checking if it satisfies the equation.

Original equation:

0.  025x + 3.75 = 1.4

Substitute x = -94:

0.  025(-94) + 3.75 = 1.4

Simplify:

-2.35 + 3.75 = 1.4

1.  4 = 1.4

Since the equation holds true, our solution x = -94 is verified.

Alternative Methods for Solving the Equation

While the step-by-step method outlined above is a standard approach, there are alternative methods we can use to solve the same equation. These methods may offer different perspectives or streamline the process, depending on individual preferences and the complexity of the equation.

Method 1: Clearing the Decimal

One approach to simplify the equation is to eliminate the decimal in the coefficient of 'x'. In our equation, 0.025x + 3.75 = 1.4, the decimal is in the term 0.025x. To clear this decimal, we can multiply both sides of the equation by a power of 10 that will shift the decimal point to the right until we have a whole number. In this case, multiplying by 1000 will do the trick.

Original equation:

0.  025x + 3.75 = 1.4

Multiply both sides by 1000:

1000(0.025x + 3.75) = 1000(1.4)

Distribute and simplify:

25x + 3750 = 1400

Now, we have an equation with whole number coefficients. We can proceed to solve for 'x' as we did before:

Subtract 3750 from both sides:
```
25x = -2350
```
Divide both sides by 25:
```
x = -94
```

We arrive at the same solution, x = -94, demonstrating the validity of this alternative method.

Method 2: Using a Calculator

In many practical situations, calculators are readily available and can significantly speed up the process of solving equations. For our equation, 0.025x + 3.75 = 1.4, we can use a calculator to perform the arithmetic operations involved in isolating 'x'.

Subtract 3.75 from both sides (using a calculator):
```
0.025x = 1.4 - 3.75 = -2.35
```
Divide both sides by 0.025 (using a calculator):
```
x = -2.35 / 0.025 = -94
```

This method highlights the efficiency of using a calculator to handle numerical computations, especially when dealing with decimals or fractions. However, it's crucial to understand the underlying algebraic principles to ensure you are applying the correct operations.

Common Mistakes to Avoid

Solving equations can sometimes be tricky, and it's easy to make mistakes if we're not careful. Here are some common pitfalls to avoid when solving equations like 0.025x + 3.75 = 1.4:

1. Incorrectly Applying Operations

A common mistake is performing operations on only one side of the equation. Remember, to maintain the equation's balance, any operation (addition, subtraction, multiplication, division) must be applied to both sides. For example, if you subtract 3.75 from the left side, you must also subtract 3.75 from the right side.

2. Arithmetic Errors

Errors in arithmetic calculations can lead to incorrect solutions. Double-check your calculations, especially when dealing with decimals or negative numbers. Using a calculator can help reduce the risk of arithmetic errors, but it's still important to understand the underlying mathematical principles.

3. Incorrect Order of Operations

When simplifying expressions, it's crucial to follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Failing to adhere to the order of operations can result in incorrect simplifications and an incorrect solution.

4. Forgetting the Sign

Pay close attention to the signs (+ or -) of numbers, especially when dealing with negative numbers. A sign error can significantly alter the solution. For example, subtracting a negative number is the same as adding its positive counterpart.

5. Not Verifying the Solution

It's always a good practice to verify your solution by substituting it back into the original equation. This helps catch any mistakes you might have made along the way. If the solution doesn't satisfy the equation, you know there's an error that needs to be corrected.

By being mindful of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in solving equations.

Applications of Solving Linear Equations

Linear equations are not just abstract mathematical concepts; they have numerous real-world applications. Understanding how to solve them is essential for various fields and everyday situations. Let's explore some examples:

1. Financial Calculations

Linear equations are used extensively in personal finance and business. For instance, calculating loan payments, determining the cost of goods with sales tax, or analyzing investment returns often involves solving linear equations. The equation 0.025x + 3.75 = 1.4 itself could represent a simplified financial scenario, such as calculating the amount of money needed to reach a specific savings goal with a certain interest rate and initial deposit.

2. Physics and Engineering

Many physical phenomena can be modeled using linear equations. For example, calculating the distance traveled at a constant speed, determining the force required to move an object, or analyzing electrical circuits often involves solving linear equations. Engineers use these equations to design structures, machines, and systems that function efficiently and safely.

3. Chemistry

Linear equations are used in chemistry to balance chemical equations, calculate the amount of reactants and products in a chemical reaction, and determine the concentration of solutions. These calculations are crucial for understanding chemical processes and designing experiments.

4. Everyday Life

Linear equations come into play in many everyday situations. For example, calculating the total cost of groceries, determining the amount of time it will take to travel a certain distance, or splitting a bill among friends can all be solved using linear equations. The ability to solve these equations empowers us to make informed decisions and solve practical problems.

Conclusion

In this comprehensive guide, we have explored the process of solving the equation 0.025x + 3.75 = 1.4. We have taken a step-by-step approach, explained alternative methods, highlighted common mistakes to avoid, and discussed real-world applications. By mastering the techniques presented in this article, you will not only be able to solve this specific equation but also gain a deeper understanding of linear equations and their significance in mathematics and beyond. Remember, practice is key to developing your problem-solving skills. So, continue to explore and challenge yourself with new equations and mathematical problems.