Problem 16 - Prealgebra

Question

Translate each sentence into an equation. Then find each number. If 5 is decreased by 3 times a number, the result is \(-4\)

Step-by-Step Solution

Answer

The number is 3.

Step 1: Understand the Problem

Identify what is given in the problem statement and what is asked. We need to translate a verbal sentence into an algebraic equation and then solve for the unknown number.

Step 2: Identify Variables

Let the unknown number be represented by the variable \( x \). This variable will be used to form our equation.

Step 3: Translate the Sentence Into an Equation

The sentence reads: 'If 5 is decreased by 3 times a number, the result is \(-4\).' This can be translated into the equation: \[ 5 - 3x = -4 \]

Step 4: Solve the Equation

We need to solve for \( x \) in the equation \( 5 - 3x = -4 \). Start by isolating the term with \( x \):1. Subtract 5 from both sides: \[ -3x = -4 - 5 \] \[ -3x = -9 \]2. Divide each side of the equation by \(-3\): \[ x = \frac{-9}{-3} \] \[ x = 3 \]

Step 5: Verify the Solution

Substitute \( x = 3 \) back into the original conditions to ensure it satisfies the equation:- Calculate \( 3 \times 3 = 9 \).- Decrease 5 by 9: \( 5 - 9 = -4 \).- Since both sides of the equation result in \(-4\), the solution \( x = 3 \) is correct.

Key Concepts

Algebraic Translation

Algebraic translation is a key step in solving word problems. It involves converting a verbal statement into a mathematical equation. This process allows you to use the power of algebra to solve for unknowns. Typically, you'll start by identifying the key elements and operations in the sentence. Then, you'll replace words with their algebraic counterparts.
For instance, in the exercise statement "If 5 is decreased by 3 times a number, the result is \(-4\)," we translate "3 times a number" into the algebraic expression \(3x\), where \(x\) is the unknown variable. The phrase "5 is decreased by" means we subtract this expression from 5, resulting in the equation \(5 - 3x = -4\).
Approaching problems with a keen insight into language will help you efficiently transform sentences into equations. Look out for phrases that signal mathematical operations, such as "sum of," "less than," or "times," to guide your translation.

Solving Equations

Once you have translated a word problem into an algebraic equation, the task is to solve it. Solving equations involves finding the value of the unknown variable that makes the equation true. For our equation \(5 - 3x = -4\), here's how you solve it step-by-step:

• Isolate the variable term: First, we want to get \(3x\) by itself. To do this, subtract 5 from both sides of the equation. This will modify it to \(-3x = -9\).
• Simplify the equation: Now, to solve for \(x\), divide each side by \(-3\). Hence, we find \(x = 3\).
• Verify your solution: Always substitute the value back into the initial equation to ensure it satisfies all conditions. Here, replacing \(x\) with 3 in \(5 - 3x\) results in \(-4\), proving our solution is correct.

Solving equations can sometimes involve multiple steps, including combining like terms or distributing factors. Practicing different types of equations will enhance your ability to recognize and apply the right methods.

Unknown Variables

In algebra, unknown variables are symbols used to represent numbers we don't yet know. They are central to formulating and solving equations, as they allow us to express mathematical relationships clearly and concisely.

• Choosing a variable: Typically, letters like \(x\), \(y\), or \(z\) are used to denote variables. In our exercise, we chose \(x\) to represent the unknown number we are solving for.
• Role of variables in equations: Variables act as placeholders that can be manipulated algebraically to find specific numeric values. By structuring an equation, you set the stage for resolving the unknowns that the variables stand for.
• Substitution and verification: Once you find a value for your variable, always substitute it back into the original problem to verify the solution's accuracy. This ensures that your found solution makes both sides of the equation equal, as demonstrated when we checked that \(5 - 3x = -4\) holds true for \(x = 3\).

Understanding how to use and solve for unknown variables is essential in algebra, as it builds the foundation for more complex mathematical problem-solving.