Problem 87 - Beginning and Intermediate Algebra

Question

Write each sentence as an equation or inequality. Use \(x\) to represent any unknown number. Thirteen minus three times a number is 13.

Step-by-Step Solution

Answer

The equation is \(13 - 3x = 13\).

Step 1: Identify the Components

In the sentence, "Thirteen minus three times a number is 13," we identify the number as the unknown. Let's represent this unknown number by \(x\). "Thirteen" and "thirteen" both refer to the same numerical value, while "three times a number" indicates multiplication of \(x\) by 3.

Step 2: Construct the Equation

Based on the components identified, "Thirteen minus three times a number" can be translated to an algebraic expression: \(13 - 3x\). The word "is" refers to equality, thus forming an equation: \(13 - 3x = 13\).

Step 3: Write the Final Equation

Now combine the algebraic expression and the equation formation from Step 2. So, the sentence translates to the equation: \(13 - 3x = 13\).

Key Concepts

Algebraic Expressions

An algebraic expression is a mathematical phrase that can include numbers, variables, and operation symbols. It is the building block for forming more complex equations and inequalities. In the context of converting sentences into equations, understanding algebraic expressions is crucial.For instance, in the exercise, "three times a number" is expressed as the algebraic expression \(3x\), where \(x\) represents the unknown number. Similarly, "Thirteen minus three times a number" forms the expression \(13 - 3x\). By breaking down the sentence into its mathematical components, we effectively create expressions that reflect the relationships described in words.When constructing algebraic expressions:

• Use variables to stand in for unknown values.
• Incorporate numbers directly as they appear in the sentence.
• Use arithmetic operators (e.g., plus, minus) to connect numbers and variables in a meaningful way.

This process helps to transition from verbal descriptions to mathematical language, facilitating further analysis or solution.

Unknown Variables

At the heart of algebra is the use of variables to represent unknown quantities. A variable, often denoted as \(x\) or another letter, is a symbol that stands in for a number that has not been specified.In our problem, the unknown number is represented by \(x\). This allows us to write a general expression that can be manipulated mathematically. Understanding how to represent unknowns is essential, as it enables the solving of equations and inequalities where the exact value is not initially given.Here are some key points about using unknown variables:

• Variables can represent any number, making them flexible in problem-solving.
• They allow for the creation of formulas and equations that can model real-world situations.
• Solving for a variable usually involves finding its value that satisfies the given equation or inequality.

By practicing how to convert sentences into equations using variables, students can develop an intuitive sense for handling unknowns and discover valuable problem-solving skills.

Equations and Inequalities

Equations and inequalities are fundamental tools in mathematics that express relationships between numbers and variables. In the exercise provided, we are tasked with translating a sentence into an equation.Once we have the algebraic expression \(13 - 3x\), the word "is" in the sentence signals an equation, leading us to write \(13 - 3x = 13\). This equation tells us that the expression on the left equals the number on the right.Here’s how equations and inequalities function:

• Equations assert that two expressions are equal, which is essential for solving problems that require finding exact values.
• Inequalities indicate a comparison and typically use symbols like \(<\), \(>\), \(\leq\), and \(\geq\) to show that expressions are not necessarily equal but have a different kind of relationship.

Understanding when to use equations versus inequalities is key. In equations, we're seeking precise solutions, while inequalities describe ranges of possible solutions. Mastering these concepts helps in formulating and solving a wide range of mathematical problems.