Problem 62 - Prealgebra

Question

Translate each sentence below into an equation, then solve the equation. The sum of 8 and \(3 x\) is 2

Step-by-Step Solution

Answer

The value of \(x\) is -2.

Step 1: Translate the Sentence

The given sentence states 'The sum of 8 and \(3x\) is 2'. In mathematical terms, the word 'sum' indicates addition. So, we translate the statement into the equation: \(8 + 3x = 2\).

Step 2: Isolate the Variable

We need to solve for \(x\). Start by isolating \(3x\) on one side. Subtract 8 from both sides of the equation: \(8 + 3x - 8 = 2 - 8\), simplifying to \(3x = -6\).

Step 3: Solve for x

Next, solve for \(x\) by dividing both sides of the equation by 3: \(\frac{3x}{3} = \frac{-6}{3}\). This simplifies to \(x = -2\).

Key Concepts

Solving Equations

To understand how to solve algebraic equations, start by knowing that an equation is a mathematical statement that shows the equality of two expressions. Solving equations involves finding the value of the unknown variable that makes the equation true. Let's break down the process with an example:

Given the equation:

• Step 1: Always look for the operation that needs to be performed to isolate the variable. In our example, the equation is \(8 + 3x = 2\). Here, the goal is to solve for \(x\), the variable.
• Step 2: First, perform inverse operations to isolate the term with the variable. Subtract 8 from both sides of the equation to eliminate the constant term \(8\) from the left side, resulting in \(3x = -6\).
• Step 3: Finally, divide both sides by 3 to solve for \(x\), giving us \(x = -2\).

Once you find the value of \(x\), plug it back into the original equation to ensure that it stays true. This method is essential for linear equations and can be adapted to solve more complex expressions.

Translating Sentences into Equations

Translating sentences into algebraic equations is a critical skill in mathematics that helps in problem-solving. The process bridges the gap between word problems and mathematical solutions. Here, let's explore how one can transform a descriptive sentence into a clear and solvable equation.

Consider the sentence: "The sum of 8 and \(3x\) is 2." Here's how you convert it:

• Identify Key Words: Words like "sum", "difference", "product", and "quotient" determine the operations. Here, "sum" indicates addition.
• Write the Equation: Replace the words with mathematical symbols. The sentence converts to \(8 + 3x = 2\).

By dissecting each part of the sentence into its mathematical components, you ensure that the equation accurately represents the statement. Practice by identifying phrases and their corresponding operations can improve your skills in translating sentences.

Mathematical Operations

Understanding mathematical operations is crucial for solving equations and tackling various math problems effectively. The basic operations include addition, subtraction, multiplication, and division. Let's see how they play a vital role in solving our example equation:

• Addition: Seen in the term \(8 + 3x\), where addition combines values into a sum.
• Subtraction: Used to isolate variables, like subtracting \(8\) from both sides in \(8 + 3x = 2\) to yield \(3x = -6\).
• Multiplication: This is in the expression \(3x\), meaning \(3\) times \(x\).
• Division: Applied in isolating the variable by dividing \(3x = -6\) by \(3\) to find \(x = -2\).

Each operation plays a distinct role in equation solving, building a foundation for more complex algebraic manipulations. Knowing when and how to apply these operations is crucial for math proficiency.