Problem 10 - Algebra A Combined Function

Question

Solve the solutions have been started for you. A 46 -foot piece of rope is cut into three pieces so that the second piece is three times as long as the first piece, and the third piece is two feet more than seven times the length of the first piece. Find the lengths of the pieces.

Step-by-Step Solution

Answer

4 feet, 12 feet, 30 feet.

Step 1: Define the Variables

Let the length of the first piece of rope be \( x \) feet. According to the problem, the second piece is three times as long, so it is \( 3x \). The third piece is two feet more than seven times the first piece, so it is \( 7x + 2 \).

Step 2: Set Up the Equation

The total length of the rope is 46 feet. We can set up the equation based on the sum of the pieces: \( x + 3x + (7x + 2) = 46 \).

Step 3: Simplify the Equation

Combine like terms in the equation: \( x + 3x + 7x + 2 = 46 \) simplifies to \( 11x + 2 = 46 \).

Step 4: Solve for x

Subtract 2 from both sides to isolate the term with \( x \): \( 11x = 44 \). Divide both sides by 11 to solve for \( x \): \( x = 4 \).

Step 5: Find the Lengths of Each Piece

The first piece is \( x = 4 \) feet. The second piece is \( 3x = 3 \times 4 = 12 \) feet. The third piece is \( 7x + 2 = 7 \times 4 + 2 = 30 \) feet.

Key Concepts

Algebraic Expressions

When solving problems with algebra, it often involves using algebraic expressions. These expressions are combinations of numbers, variables, and operation symbols that stand in for values. In our problem involving the rope, expressions like \( x \), \( 3x \), and \( 7x + 2 \) are used. They represent the lengths of the pieces of rope.

Understanding how to construct these expressions is crucial because they allow us to represent unknown quantities and relate them to each other using known operations. They form the backbone of setting up equations that can be solved to find the unknown values, much like we did here with the lengths of the pieces of rope.

Variable Definition

In algebra, variables are symbols used to represent unknown values. Here, the variable is \( x \), which we defined as the length of the first piece of rope. This choice of variable is crucial as it simplifies our problem.

By defining the length of the first piece as \( x \), we can express the other pieces in terms of \( x \). This step is key in solving the problem because it converts a word problem into a math problem, making it easier to handle.

• First piece: \( x \)
• Second piece: \( 3x \)
• Third piece: \( 7x + 2 \)

Clear variable definition allows us to solve complex problems methodically.

Equation Setup

Setting up equations from a word problem is a crucial skill in algebra. After defining the variables, the next step is to express the relationships between them using an equation. For our problem, the total length of the rope is given as 46 feet.

We use this information to set up the equation: \( x + 3x + (7x + 2) = 46 \). This equation incorporates all parts of the rope in terms of the variable \( x \), ensuring that we can find the solution using algebraic methods.

Translating word problems into equations requires careful reading and understanding of the problem, but it is always about converting known relationships and constraints into the language of math.

Simplifying Equations

Simplifying equations is essential to solving them. This process involves combining like terms and performing basic operations to make the equation easier to solve.

In our equation \( x + 3x + 7x + 2 = 46 \), we simplify it by combining like terms to get \( 11x + 2 = 46 \). This step reduces the complexity of the equation, making it easier to solve.

• Combine like terms: \( x + 3x + 7x = 11x \)
• Simplified equation: \( 11x + 2 = 46 \)

Simplification helps in clearly isolating the variable, ultimately leading to finding the solution for \( x \). Being proficient in this skill is crucial as it is regularly used not just in algebra but throughout mathematics.