Problem 9 - Prealgebra

Question

Translate each sentence into an equation. Then find each number. Eight less than ten times a number is 82 .

Step-by-Step Solution

Answer

The number is 9.

Step 1: Understanding the Problem

The sentence "Eight less than ten times a number is 82" needs to be translated into a mathematical equation. Start by identifying the unknown, which we'll call \(x\), representing the number we're looking for.

Step 2: Translating Words into Equation

The phrase "ten times a number" can be represented as \(10x\). The phrase "eight less than" implies subtraction of 8 from something. Therefore, the equation becomes: \(10x - 8 = 82\).

Step 3: Isolating the Variable

To solve for \(x\), we first add 8 to both sides of the equation to eliminate the -8. This gives us: \(10x = 82 + 8\), which simplifies to \(10x = 90\).

Step 4: Solving for the Number

Now divide both sides by 10 to solve for \(x\). This gives: \(x = \frac{90}{10}\), which simplifies to \(x = 9\).

Key Concepts

Equation Translation

When you come across a word problem, translating that text into an equation is the first step. Imagine you're translating a sentence from English into a mathematical language. Start by identifying the unknown value in the problem, which is often represented by a variable, such as \( x \). In this exercise, we're told that we are dealing with "ten times a number" and "eight less than" this product, set equal to 82.

• "Ten times a number": This translates to \( 10x \) with \( x \) being our unknown number.
• "Eight less than": This suggests a subtraction of 8 from another quantity. Here, it pertains to \( 10x - 8 \).
• "Is": In mathematical language, this word usually indicates an equals sign \( = \).

Thus, the sentence transforms into this equation: \( 10x - 8 = 82 \). Understanding this translation process is crucial in tackling more complex problems later on.

Solving Equations

Once you have an equation, the task is to solve it, meaning you'll find the value of the variable that makes the equation true. Solving an equation is like unraveling a puzzle; you need to carefully follow certain steps to isolate your variable. This involves operations such as addition, subtraction, multiplication, and division.First up, we need to "undo" the subtraction part of the equation:

• To remove \(-8\), we add 8 to both sides. This helps to keep the balance, giving us \( 10x = 90 \).
• Next, to isolate \( x \), divide both sides by 10. This division step helps simplify the equation further to get \( x = 9 \).

These steps are all about maintaining equality and transforming the equation into a simpler form. Practice makes perfect here. As you get more familiar, solving equations will become second nature.

Basic Algebra Concepts

Diving into prealgebra involves getting to know the essential building blocks of algebra. The key is understanding variables, like \( x \), which are placeholders for numbers we aim to find. The basic operations of addition, subtraction, multiplication, and division are tools used to manipulate these variables and solve equations.A strong grasp of these foundational concepts allows you to tackle problems of increasing complexity. Some fundamental aspects include:

• Variables: Bring flexibility into equations and represent unknown numbers.
• Operations: The building blocks—adding, subtracting, multiplying, and dividing—enable you to rearrange and solve equations.
• Equality: Always perform the same operation on both sides of the equation to maintain balance.

Understanding these basics clears the path for more advanced topics like functions, expressions, and more intricate algebraic equations. As you progress, remember that the principles remain the same, only the complexity increases.