Problem 64 - Prealgebra

Question

Translate each sentence below into an equation, then solve the equation. The difference of 9 and \(6 x\) is 21

Step-by-Step Solution

Answer

The solution is \( x = -2 \).

Step 1: Translate the Sentence into an Equation

We start by translating the sentence "The difference of 9 and \( 6x \) is 21" into a mathematical equation. The word "difference" indicates a subtraction operation. Therefore, the sentence can be translated to the equation: \( 9 - 6x = 21 \).

Step 2: Isolate the Variable Term

Our goal is to solve for \( x \). We begin by isolating the term involving \( x \) on one side of the equation. We subtract 9 from both sides of the equation to achieve this: \( 9 - 6x - 9 = 21 - 9 \), simplifying to \( -6x = 12 \).

Step 3: Solve for x

With the equation \( -6x = 12 \), we solve for \( x \) by dividing both sides by -6: \( x = \frac{12}{-6} \). This simplifies to \( x = -2 \).

Key Concepts

Solving Equations

Equations are sentences made with numbers and symbols, and solving them means finding the value of the unknown quantity, often represented by a variable. In this exercise, we needed to translate a sentence into a mathematical equation and solve for the variable. The sentence told us that the difference of 9 and \(6x\) equals 21. The word “difference” signifies subtraction. Thus, we wrote it as \(9 - 6x = 21\). To solve an equation like this, you need to perform operations that simplify it until the variable, in this case \(x\), is by itself on one side of the equation. What we did here was reversing the operations, exactly like turning blocks backward to find where they first began. The solution uses a combination of addition, subtraction, multiplication, or division. In this situation, the equation needed both subtraction and division.

Isolating Variables

Isolating the variable is a key step in solving an equation. In mathematical terms, isolating a variable means getting the variable alone on one side of the equation. This process helps us easily identify the value of the variable.Let’s break down how we isolate \(x\) in the equation \(9 - 6x = 21\). Our goal is to remove everything else surrounding the term \(6x\).

• First, we subtracted 9 from both sides, calculated as \(9 - 9\) and \(21 - 9\), simplifying the equation to \(-6x = 12\).
• The minus sign in front of \(6x\) meant we still needed to deal with multiplication by \(-6\). Thus, we divided every side of the equation by \(-6\), simplifying it to \(x = -2\).

Practicing isolation helps with understanding the core concept of algebra, allowing variables to literally stand alone and reveal their value.

Prealgebra Concepts

Prealgebra is like the alphabet of math. It involves basic math skills that prepare students for algebra. This includes basic operations, understanding equations, and learning how variables work. This exercise involved:

• Identifying and understanding mathematical expressions from sentences.
• Working with operations such as subtraction and division.
• Learning how to manipulate equations to solve for unknown variables.

Recognizing these basic operations is essential. Prealgebra effectively lays the groundwork for solving more complex algebraic problems in the future. It's important because it builds a foundation of problem-solving skills that are crucial as one advances in mathematics.