Problem 96 - Introductory and Intermediate Algebra for College Students 4th

Question

Explain how to clear decimals in a linear equation.

Step-by-Step Solution

Answer

To clear decimals from a linear equation, the equation is multiplied by a power of 10 that corresponds to the decimal with the most digits after the point. This transforms the equation into an equivalent one but without decimals, making it simpler to solve.

Step 1: Understand the task

Clearly understand the task at hand which, in this case, is to eliminate decimals from the given linear equation.

Step 2: Identify the decimal with the most numbers after the point

In the equation, identity the decimal that has the most number of digits after the decimal point. This is important because all decimals in the equation need to be cleared at once. Thus, to do this, one needs to find the decimal with most digits after the decimal point.

Step 3: Clear decimals

To clear decimals from each term of the equation, multiply each term of the equation by 10 raised to the power equal to the number of digits in the largest decimal identified in Step 2. This ensures that each term is free of decimals.

Step 4: Solving the cleared equation

After clearing the decimals, proceed with solving the equation as a simple linear equation. Using this method helps to avoid unnecessary mistakes that can happen when working with decimals.

Key Concepts

Decimals

Decimals are a way to express fractions or parts of whole numbers using a period or a point. In linear equations, decimals often represent precise numbers that need to be managed carefully. When solving equations, working with decimals can be tricky, as calculations can become cumbersome. To simplify, you can eliminate decimals by finding a common factor, usually a power of 10, to clear the equation from decimal representation. This helps in making the problem easier to understand and solve. Keep in mind:

• Decimals ensure precise value representation, but can complicate calculations.
• Clearing decimals by multiplying can streamline solving processes.
• Converting all decimals to whole numbers helps reduce the chance of error.

Being comfortable with decimals increases your accuracy in solving linear equations.

Equation Clearance

Clearing an equation means transforming it to a form that is simpler to manage, often turning all terms into integers. This technique is especially useful when dealing with decimals.
To clear decimals from a linear equation:

• Identify the term with the most decimal places.
• Multiply every term in the equation by a power of 10 that matches this number of decimal places.

This multiplication effectively shifts the decimal point, converting all coefficients into integers. For example, if the maximum number of decimal places is 2, multiply each term by 100.
This process simplifies the equation without affecting the balance between both sides, keeping the equation's core properties intact, readying it for straightforward solution methods.

Problem Solving Steps

Solving a linear equation, especially when decimals are involved, requires a series of methodical steps to ensure accuracy.
Here's an approach to tackle such problems:

• Understand the Problem: Clearly define what needs to be solved, setting a strategy based on the presence of decimals.
• Identify Decimal Positions: Locate the term with the most decimal places for effective clearance.
• Clear Decimals: Multiply the entire equation by an appropriate power of ten.
• Solve the Equation: With decimals removed, solve the linear equation using standard algebraic methods.

Ensure every step is worked through carefully to maintain equation balance and accuracy, preventing errors often encountered when first tackling equations with decimals.