Problem 83 - Algebra A Combined Function

Question

Write each algebraic expression described. To convert from feet to inches, we multiply by 12 . For example, the number of inches in 2 feet is \(12 \cdot 2\) inches. If one board has a length of \((x+2)\) fee and a second board has a length of \((3 x-1)\) inches, express their total length in inches as an algebraic expression.

Step-by-Step Solution

Answer

The total length in inches is \(15x + 23\).

Step 1: Convert Feet to Inches

The first board is measured in feet. To express its length in inches, we multiply by 12. Thus, the length of the first board is \(12 \cdot (x + 2)\) inches.

Step 2: Identify the Expression for the Second Board

The length of the second board is already given in inches, expressed as \(3x - 1\). No conversion is needed for this board.

Step 3: Combine Both Expressions

To find the total length of both boards in inches, we need to add the expressions from Step 1 and Step 2. This gives us: \(12(x + 2) + (3x - 1)\).

Step 4: Simplify the Expression

Distribute the 12 in the first term: \(12x + 24\). Now, combine like terms from the full expression: \(12x + 24 + 3x - 1 = 15x + 23\).

Key Concepts

Algebraic Expressions

Algebraic expressions are a powerful tool in mathematics. They are mathematical sentences that use numbers, variables, and operations to describe real-world situations or solve problems. For example, the expression \((x + 2)\) represents a quantity that is two more than \(x\), while \(3x - 1\) indicates an amount where one is subtracted from three times \(x\).
Understanding algebraic expressions is essential because they help us model and solve problems systematically. They're composed of terms, which can be constants, variables, or a mix of both. By learning how to write and manipulate these expressions, we gain a deeper understanding of how quantities relate and change. When dealing with algebraic expressions, always remember:

• Identify the variables used and what they represent.
• Pay attention to the operations performed on these variables.

Whether you're converting measurements or finding total quantities, expressions are your key to unlocking solutions.

Unit Conversion

Unit conversion is the process of converting a quantity from one measurement unit to another. This skill is handy in everyday scenarios, especially in tasks involving measurements like length. In our exercise, we convert feet to inches.
To convert a measurement given in feet to inches, multiply the quantity by 12 because there are 12 inches in a foot. For the expression involving the length \((x+2)\) feet, multiplying by 12 gives its length in inches as \(12(x+2)\).
Remember these tips for unit conversion:

• Always know the conversion factor (e.g., 12 inches in a foot).
• Multiply or divide, as necessary, to perform the conversion.

Conversions simplify calculations by maintaining consistent units throughout a problem, reducing confusion and errors.

Simplifying Expressions

Simplifying expressions involves reducing them to their simplest form, making them easier to interpret or solve. Once you have an algebraic expression, the next step often includes simplifying it by performing operations and combining terms.
In our example, after converting the board lengths to inches, we arrived at the expression \(12(x + 2) + (3x - 1)\). The process of simplifying includes:

• Distributing multiplication over addition, such as transforming \(12(x + 2)\) into \(12x + 24\).
• Combining like terms to consolidate the expression into a manageable form, resulting in \(15x + 23\).

Simplifying expressions is a crucial step in algebra to ensure your results are clear and concise. Always aim for the simplest form of an expression when solving or presenting solutions.

Like Terms

Like terms are a fundamental concept when working with algebraic expressions. They are terms within an expression that have the same variables raised to the same power, differing only by coefficients. Identifying and combining like terms simplifies expressions.
For instance, in the expression \(12x + 24 + 3x - 1\), you can spot the like terms \(12x\) and \(3x\). These terms both have the variable \(x\), making them "like." To combine them, you simply add their coefficients: \(12 + 3 = 15\), giving you \(15x\).
Here are some tips to effectively work with like terms:

• Always ensure the terms have identical variables and exponents before combining.
• Only the coefficients of these terms are combined; variable parts remain unchanged.

Mastering like terms allows you to efficiently simplify expressions and solve algebraic equations.