Problem 40 - Algebra 1

Question

Multiply the equation by a power of 10 to write an equivalent equation with integer coefficients. $$ 2.5 x+0.7=4.6-1.3 x $$

Step-by-Step Solution

Answer

The equivalent equation with integer coefficients is \(25x + 7 = 46 - 13x\)

Step 1 Title: Identify the Decimals

Firstly, look at the equation \(2.5x + 0.7 = 4.6 - 1.3x\). Identify the decimal places in the coefficients.

Step 2 Title: Determine the power of 10

Next, determine the power of 10 that will eliminate all decimals. Here, the highest number of decimal places is 1. Therefore, multiplying the entire equation by \(10^1 = 10\) will convert the equation to integer coefficients.

Step 3 Title: Multiply the equation

Now, multiply every term in the equation by 10. So the equation becomes: \(10 * 2.5x + 10 * 0.7 = 10 * 4.6 - 10 * 1.3x\), which simplifies to \(25x + 7 = 46 - 13x\)

Key Concepts

Multiply by Powers of Ten

When working with equations that have decimal coefficients, one effective strategy is to multiply each term by a power of ten. This is done to remove decimals and achieve integer coefficients, which are typically easier to work with. The power of ten you choose depends on the number of decimal places in the coefficients. In this exercise, the maximum number of decimal places in any term is one, meaning you need to multiply the entire equation by 10 (or $10^1$) to eliminate the decimals.

• Multiply each term by 10.
• The decimals (like 0.7) are converted into whole numbers (like 7).
• This transformation simplifies the arithmetic and makes the equation more manageable.

Understanding when and how to multiply by powers of ten is essential for handling decimals in algebraic equations. This process also involves recognizing the decimal placement and correctly calculating the power of ten needed.

Transforming Equations

Transforming equations involves changing their form but not their value. This process helps make equations easier to solve or understand. In this exercise, transforming the equation by multiplying each term by a power of ten is a key step. By doing so, we remove the inconvenient decimal fractions and achieve a more straightforward equation with integer coefficients.

• The original equation is multiplied by 10 to clear decimals.
• The transformed version is $25x + 7 = 46 - 13x$, which is as valid as the original but easier to handle computationally.
• This method preserves the equality, meaning both sides still represent the same relationship as before.

This transformation is crucial for solving equations efficiently and accurately. It becomes especially helpful when working with more complex algebraic solutions.

Algebraic Manipulation

Algebraic manipulation is the skillful use of operations to change and rearrange equations. It is fundamental to finding solutions and is used extensively after transforming equations with powers of ten. Once you have integer coefficients, you can apply various algebraic techniques to simplify or solve the equation. For instance, you may add, subtract, or multiply terms to isolate variables.

• Move terms strategically across the equality sign.
• Combine like terms whenever possible.
• Use the inverse operations to solve for variables (e.g., if you have $25x = 46 - 13x$, you can move all $x$ terms to one side to solve for $x$).

These manipulations help break down complex problems into manageable steps and find the solution systematically. Mastering algebraic manipulation is necessary for tackling diverse equations and mathematical challenges.