Problem 43 - Algebra 1

Question

Multiply the equation by a power of 10 to write an equivalent equation with integer coefficients. $$ 4.5 n-0.375=0.75 n+2.0 $$

Step-by-Step Solution

Answer

Multiplying the equation \(4.5n - 0.375 = 0.75n + 2.0\) by 1000 results in the equivalent equation with integer coefficients: \(4500n - 375 = 750n + 2000\).

Step 1: Identifying the smallest decimal place

Analyze the equation \(4.5n - 0.375 = 0.75n + 2.0\). To transform all the decimals into whole numbers, we need to identify the decimal place with the most number of digits after it. In this case, we have a three decimal places in -0.375.

Step 2: Choosing the correct power of 10

Every term in the equation will be multiplied by the power of 10 which corresponds to the biggest number of digits after the decimal point. In this case, that is three, so we will multiply every term by \(10^3\) or 1000.

Step 3: Multiplying the equation by \(10^3\)

Multiply each term in the equation by 1000 to get the new equation: \(1000*4.5n - 1000*0.375 = 1000*0.75n + 1000*2.0\), which simplifies to: \(4500n - 375 = 750n + 2000\).

Key Concepts

Multiplying Equations by a Power of 10

When dealing with equations containing decimals, it can be easier to work with them after eliminating the decimal points. This can be achieved by multipling the entire equation by a power of 10. Essentially, for every decimal place you want to move to the right (to turn a decimal into a whole number), you multiply by ten. If a decimal has three places, as in 0.375, you'll multiply by 1000, or 10^3, to shift the decimal three places to the right, transforming it into 375.

When you apply this to each term in the equation, you ensure the equation remains in balance—because what you do to one side, you must do to the other. This method creates an equivalent equation with integer coefficients, which are often simpler to work with when solving algebraic equations.

Transforming Decimal Coefficients to Integers

Decimal coefficients in algebra can complicate matters. To streamline the solving process, convert these coefficients to whole numbers by identifying the term with the highest number of decimal places and multiplying each term by a corresponding power of ten to 'move' the decimal point to the right until you have a whole number. This transformation allows for easier manipulation of the equation, leading to a clearer path to finding the value of the variable in question.

• If a coefficient has one decimal place, multiply by 10^1 (10).
• If it has two decimal places, multiply by 10^2 (100).
• For three decimal places, use 10^3 (1000), and so on.

Through this process, what might initially appear as a complex equation with decimals becomes a more approachable problem with integer coefficients.

Simplifying Algebraic Equations

Simplification is a key skill in algebra that involves reducing the complexity of an equation, making it easier to solve. Integral coefficients are usually a good starting point for simplification. After multiplying by a power of 10 and eliminating decimals, the equation can often be further simplified by combining like terms or by moving terms across the equals sign such that each variable and its coefficient become clearer.

Combining Like Terms

For instance, if an equation has terms 4500n and -750n on different sides, you could subtract 750n from both sides of the equation to combine them into a single term. This way, simplification facilitates the isolation of variables and moves you closer to solving the equation.

Decimal Place Value Analysis

Place value is crucial when working with decimals. Understanding that each move to the right in a decimal signifies a decrease in value by a factor of 10 is fundamental. For instance, in the number 0.375, the 3 is in the tenths place, the 7 in the hundredths place, and the 5 in the thousandths place.

When you multiply an equation by a power of 10, you're using decimal place value analysis to ensure every decimal is moved the right number of places. The same principle applies in reverse when dividing by powers of 10. This understanding is crucial for not just solving equations with decimals, but also for grasping how numbers themselves are structured and how their values can be manipulated in an algebraic context.