Problem 26 - Algebra 1

Question

Translate the verbal sentence into an equation. 3 more than a number is 5

Step-by-Step Solution

Answer

The algebraic expression for the verbal sentence '3 more than a number is 5' is \(x + 3 = 5\).

Step 1: Identify the variable

In this statement, 'a number' is the variable. Let's denote this number as \(x.\)

Step 2: Translate verbal phrases

The phrase '3 more than a number' suggests that 3 is being added to our variable \(x\). Therefore, it can be translated to \(x + 3\). The term 'is' suggests equality and can be translated to '='. So, the verbal sentence '3 more than a number is 5' translates to the algebraic expression \(x + 3 = 5\).

Key Concepts

Algebraic Expressions

Algebraic expressions are the cornerstone of algebra. They combine numbers and variables using arithmetic operations: addition (+), subtraction (−), multiplication (×), and division (÷). Unlike equations, these expressions don't necessarily have an equality sign; they simply represent a value or relationship.

An expression can be as simple as a single variable or as complex as a combination of multiple operations and groupings. For example, in the exercise '3 more than a number,' the expression '3 more than' translates to +3 being added to a certain variable.

Understanding Components

Variables are like containers for numbers we might not know yet, and constants are specific, unchanging numbers. Operations tell us what to do with these numbers and variables. Together, they form the language of algebra, allowing us to describe relationships and solve problems.

• Variables: symbols like \(x\), \(y\), or \(z\) representing unknown quantities
• Constants: known values, like the number 3 in our exercise
• Operations: how we combine variables and constants (e.g., addition, subtraction)

Equations in Algebra

Equations are the bread and butter of algebra. They consist of two algebraic expressions set equal to each other with the use of an '=' sign. The main goal when working with equations is usually to find the value of the unknown variables that make the equation true.

In the provided exercise, we encounter the equation \(x + 3 = 5\). This equation tells us that when we add 3 to some unknown number \(x\), we get 5.

Unlocking Equations

To solve an equation, we perform operations to isolate the variable, giving us insight into its possible value(s). In the case of \(x + 3 = 5\), subtracting 3 from both sides of the equation would reveal the value of \(x\). Equations can be simple, like the one we have, or they can be complex with variables on both sides and require more steps to solve.

• Single-Step Equations: can be solved with one operation (like our example)
• Multi-Step Equations: require more than one operation to isolate the variable

Variable Identification

Identifying the variable is a critical first step in translating a verbal sentence into an equation. The variable represents an unknown quantity that we are trying to find. It's often a letter like \(x\), \(y\), or \(z\), which stands in for a number.

In the earlier exercise, the variable was identified through the phrase 'a number,' which doesn't tell us what the number actually is, hence the use of a variable \(x\). Once you've identified your variable, you can construct the rest of your equation around it.

Choosing a Variable

When choosing a symbol for your variable, you can use any letter that makes sense to you or is indicated by the problem. However, it's common to start with \(x\) and then use \(y\), \(z\), and so on, if more variables are needed.

• Context: Sometimes, variables are chosen based on context (e.g., \(t\) for time)
• Clarity: Choosing distinctive variables helps prevent confusion in more complex problems.