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Chapter 2: Problem 80

For the equation \(0.35 x+2.1=3.85\), how do you clear the decimal?

Short Answer

Expert verified
Multiply each term by 100 to clear the decimals: \(35x + 210 = 385\).

Step by step solution

01

- Understand the Equation

The given equation is \(0.35x + 2.1 = 3.85\). To clear the decimals, we should multiply every term by the same power of 10 that will convert all terms into whole numbers.
02

- Identify the Multiplier

Determine the smallest power of 10 that can be used to convert the decimals to whole numbers. In this case, the highest decimal place is two places (hundredths place). Therefore, we should multiply every term by 100.
03

- Multiply Each Term

Multiply each term in the equation \(0.35x + 2.1 = 3.85\) by 100. The equation becomes \100 * 0.35x + 100 * 2.1 = 100 * 3.85\.
04

- Simplify the Equation

Simplify each term after multiplying: \(100 * 0.35x = 35x\), \(100 * 2.1 = 210\), \(100 * 3.85 = 385\). The simplified equation is \35x + 210 = 385\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

clearing decimals
When solving equations with decimal coefficients, it's often easier to work with whole numbers. To clear the decimals, identify the highest number of decimal places in the equation. For example, in the equation \(0.35x + 2.1 = 3.85\), the highest decimal place is the hundredths place (two decimal places). Multiply every term by 100 to convert all terms into whole numbers. This process ensures that you no longer have decimals in the equation, making it simpler to solve.
multiplying equations
Multiplying every term in an equation by the same number does not change the equality of the equation. It simply scales each term proportionally. In our example, multiplying \(0.35x + 2.1 = 3.85\) by 100, we get \(100 \times 0.35x + 100 \times 2.1 = 100 \times 3.85\). This step converts the equation to \(35x + 210 = 385\). It's crucial to multiply every term to preserve the balance of the equation.
simplifying equations
Once an equation has been scaled up to remove decimals, its terms may be simplified. In our worked equation, we have \(100 \times 0.35x\), \(100 \times 2.1\), and \(100 \times 3.85\). Simplifying these terms gives us the whole numbers: 35x, 210, and 385. Therefore, the equation \(35x + 210 = 385\) is much easier to handle and can be solved using basic algebraic techniques.
whole numbers
Working with whole numbers in equations usually simplifies calculations and reduces errors. In our example, turning \(0.35x + 2.1 = 3.85\) into \(35x + 210 = 385\) makes the terms easier to work with. Dealing with whole numbers not only simplifies the solving process but also helps in accurate checking and verification of the solution.

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