Problem 11 - Algebra A Combined Function

Question

A 40 -inch board is to be cut into three pieces so that the second piece is twice as long as the first piece and the third piece is 5 times as long as the first piece. If \(x\) represents the length of the first piece, find the lengths of all three pieces.

Step-by-Step Solution

Answer

The lengths of the pieces are 5, 10, and 25 inches.

Step 1: Define the Variables

Let \( x \) be the length of the first piece of the board. According to the problem, the second piece is twice the length of the first piece, and the third piece is five times the length of the first piece.

Step 2: Express Lengths of Each Piece

Based on the variable \( x \), express the lengths of each board piece:- First piece: \( x \)- Second piece: \( 2x \)- Third piece: \( 5x \)

Step 3: Set Up the Equation

Since the lengths of the three pieces together must equal the original length of the board (40 inches), set up the equation: \[ x + 2x + 5x = 40 \]

Step 4: Solve the Equation

Simplify the equation to find \( x \):- Combine like terms: \( 8x = 40 \)- Solve for \( x \): \[ x = \frac{40}{8} = 5 \]

Step 5: Calculate Each Piece's Length

Using the value of \( x \) calculated, determine the lengths of each piece:- First piece: \( x = 5 \) inches- Second piece: \( 2x = 2 \times 5 = 10 \) inches- Third piece: \( 5x = 5 \times 5 = 25 \) inches

Key Concepts

Linear Equations

Linear equations are fundamental in algebra and involve expressions set equal to a specific value. In this exercise, the board's total length of 40 inches is split into three pieces. We express this as the linear equation \( x + 2x + 5x = 40 \). Linear equations are called so because their graph forms a straight line if plotted. They only include variables raised to the power of one.

Solving a linear equation typically involves simplifying and reducing the equation to find the value of the variable. Here, by combining like terms, we simplify the equation to \( 8x = 40 \). Our goal is to isolate the variable \( x \) on one side, which is a basic step in solving such equations.

Variables and Expressions

Variables and expressions are essential tools in algebra. A variable is a symbol, often a letter like \( x \), used to represent unknown values. In the given problem, \( x \) represents the length of the first board piece. The remaining lengths are expressed in terms of \( x \) as well: \( 2x \) for the second piece and \( 5x \) for the third piece.

Expressions combine variables, numbers, and operations like addition or multiplication. In our problem, each expression corresponds to the length of a piece of the board. Expressions help translate words from a problem into mathematical phrases. This step is critical for setting up equations that will allow us to find solutions. Understanding how to construct expressions correctly ensures you can solve problems accurately.

Problem-Solving Steps

Problem-solving in algebra involves several clear steps that can be applied to various problems. First, define the variables that represent the unknowns in the problem. Next, convert the text of the problem into mathematical expressions. This involves understanding the relationships described in the problem.

In this exercise, once the expressions for each board piece are formed, the next step is setting up the equation: \( x + 2x + 5x = 40 \). Solving this equation involves simplification, which leads to \( 8x = 40 \). Divide both sides by 8 to find \( x = 5 \). Finally, use this value to find the lengths of each piece. By structuring your approach with these steps, you clearly map out a path to the solution.