Eliminating Decimals In Equations A Step By Step Guide

To effectively solve equations involving decimals, a crucial first step is often eliminating these decimals. This simplifies the equation and makes it easier to manipulate. In this comprehensive guide, we will delve into the equation $-m + 0.02 + 2.1m = -1.45 - 4.81m$ and explore the optimal number to multiply each term by in order to eliminate the decimals. Understanding this process is fundamental for anyone studying algebra or related mathematical fields. We will not only identify the correct number but also explain the reasoning behind it, ensuring a solid grasp of the underlying mathematical principles.

Understanding Decimal Elimination

Before diving into the specific equation, let's establish the basic principle of eliminating decimals in an equation. The goal is to transform the equation into one with integer coefficients, making it easier to solve using standard algebraic techniques. Decimals represent fractions with denominators that are powers of 10, such as 0.1 (tenths), 0.01 (hundredths), and 0.001 (thousandths). To eliminate decimals, we need to multiply the entire equation by a power of 10 that will shift the decimal point enough places to the right to convert all decimal numbers into integers. The choice of the power of 10 depends on the decimal with the most digits after the decimal point. This foundational understanding is crucial for tackling various algebraic problems efficiently.

Identifying the Decimal Places

In our equation, $-m + 0.02 + 2.1m = -1.45 - 4.81m$, we need to carefully examine each term containing a decimal. We have 0.02, 2.1, -1.45, and -4.81. The number 0.02 has two decimal places (hundredths), 2.1 has one decimal place (tenths), -1.45 has two decimal places (hundredths), and -4.81 also has two decimal places (hundredths). To eliminate all the decimals, we need to consider the term with the most decimal places, which in this case is two decimal places (hundredths).

The Role of Powers of 10

Powers of 10 are essential in manipulating decimal numbers. Multiplying by 10 shifts the decimal point one place to the right, multiplying by 100 shifts it two places, multiplying by 1000 shifts it three places, and so on. To eliminate decimals, we choose the power of 10 that corresponds to the maximum number of decimal places present in the equation. For example, if the maximum number of decimal places is two, we multiply by 100; if it's three, we multiply by 1000, and so forth. This ensures that every decimal term is converted into an integer. Understanding this mechanism is key to successfully clearing decimals from any equation.

Determining the Correct Multiplier for the Equation

Now that we've identified the number of decimal places, we can determine the correct multiplier for our specific equation: $-m + 0.02 + 2.1m = -1.45 - 4.81m$. As we noted earlier, the maximum number of decimal places in this equation is two (present in 0.02, -1.45, and -4.81). Therefore, to eliminate all decimals, we need to multiply each term of the equation by 100. This will shift the decimal point two places to the right in each term, converting all decimal numbers into integers.

Step-by-Step Multiplication

Let's walk through the multiplication process step by step:

1. Multiply -m by 100: 100 * -m = -100m
2. Multiply 0.02 by 100: 100 * 0.02 = 2
3. Multiply 2.1m by 100: 100 * 2.1m = 210m
4. Multiply -1.45 by 100: 100 * -1.45 = -145
5. Multiply -4.81m by 100: 100 * -4.81m = -481m

After multiplying each term by 100, our equation becomes: -100m + 2 + 210m = -145 - 481m. This equation is now free of decimals and can be solved using standard algebraic techniques. This process illustrates how effectively multiplying by the correct power of 10 can simplify equations.

Why Other Options Are Incorrect

To further solidify our understanding, let's examine why the other options provided (0.01, 0.1, and 10) are not suitable for eliminating decimals in this equation:

• 0. 01: Multiplying by 0.01 would shift the decimal point two places to the left, making the numbers even smaller and introducing more decimals rather than eliminating them.
• 0. 1: Multiplying by 0.1 would shift the decimal point one place to the left, which would not eliminate all decimals, especially those with two decimal places.
• 10: Multiplying by 10 would shift the decimal point one place to the right. While this would eliminate the decimal in 2.1, it would not eliminate the decimals in 0.02, -1.45, and -4.81, which have two decimal places. This highlights the importance of choosing the multiplier based on the maximum number of decimal places.

Solving the Decimal-Free Equation

Having successfully eliminated the decimals, we now have the simplified equation: -100m + 2 + 210m = -145 - 481m. We can now solve for m using standard algebraic steps.

Combining Like Terms

The first step in solving the equation is to combine like terms on both sides. On the left side, we have -100m and 210m, which combine to give 110m. So the left side becomes 110m + 2. The equation now looks like this: 110m + 2 = -145 - 481m.

Isolating the Variable

Next, we want to isolate the variable m. We can do this by adding 481m to both sides of the equation. This gives us 110m + 481m + 2 = -145. Combining the m terms on the left side, we get 591m + 2 = -145.

Further Isolation

To continue isolating m, we subtract 2 from both sides of the equation: 591m = -145 - 2, which simplifies to 591m = -147.

Solving for m

Finally, we divide both sides of the equation by 591 to solve for m: m = -147 / 591. This fraction can be simplified by dividing both the numerator and denominator by 3, resulting in m = -49 / 197. Thus, the solution to the equation is m = -49/197. This demonstrates the complete process of solving an equation after eliminating decimals, reinforcing the importance of the initial step of clearing decimals for easier computation.

Conclusion: Mastering Decimal Elimination

In conclusion, to eliminate decimals in the equation $-m + 0.02 + 2.1m = -1.45 - 4.81m$, we need to multiply each term by 100. This is because the maximum number of decimal places in the equation is two. Understanding how to eliminate decimals is a fundamental skill in algebra and simplifies the process of solving equations. By multiplying by the appropriate power of 10, we convert decimal numbers into integers, making the equation easier to manipulate and solve. This guide has not only provided the answer but also walked through the reasoning and the subsequent steps to solve the equation, ensuring a comprehensive understanding of the topic. Mastering this skill will undoubtedly enhance your algebraic proficiency and problem-solving abilities.