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Chapter 2: Problem 35

A 17 -foot piece of string is cut into two pieces so that the longer piece is 2 feet longer than twice the length of the shorter piece. Find the lengths of both pieces.

Short Answer

Expert verified
The shorter piece is 5 feet, and the longer piece is 12 feet.

Step by step solution

01

Define Variables

Let's define the variables to make the problem simpler. Let \( x \) be the length of the shorter piece of string.
02

Express Longer Piece in Terms of x

According to the problem, the longer piece is 2 feet longer than twice the length of the shorter piece. This can be expressed as \( 2x + 2 \).
03

Set Up the Equation

The total length of both pieces is 17 feet. Therefore, we can set up the equation:\[x + (2x + 2) = 17\]
04

Simplify the Equation

Simplify the equation:\[x + 2x + 2 = 17\]Combine like terms:\[3x + 2 = 17\]
05

Solve for x

Subtract 2 from both sides:\[3x = 15\]Divide both sides by 3 to solve for \( x \):\[x = 5\]
06

Find the Length of Longer Piece

Now substitute \( x = 5 \) back into the equation for the longer piece:\[2x + 2 = 2(5) + 2 = 10 + 2 = 12\]
07

Verify the Solution

Check if the sum of both pieces adds up to 17:\[5 + 12 = 17\]Since the sum is correct, the solution is verified.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Linear Equations
Linear equations are a fundamental concept in algebra and are used to find unknown values. These equations typically form a straight line when graphed.
To solve a linear equation, follow a systematic approach:
  • Identify all terms in the equation.
  • Make sure all variables are on one side and constants on the other.
  • Simplify and then solve for the variable.
In the string-cutting exercise, we used the equation:\[x + (2x + 2) = 17\]
This equation sums the lengths of two pieces of string. With algebraic manipulation, we found the unknown value for the shorter string length, which is a crucial step in solving linear equations.
Variable Definition
Variable definition is about choosing a symbol, often a letter, to represent an unknown quantity in a problem. This makes it easier to work with and ultimately solve the problem.
In our exercise, we defined the variable \(x\) to represent the length of the shorter piece of string.
  • Choosing the variable early clarifies what you are looking for.
  • It allows you to express other quantities in the problem in terms of this variable.
By understanding variable definitions, you can clearly set the stage for solving the rest of the equation. Here, with \(x\) established, we could then express the longer string piece in terms of \(x\).
Equation Simplification
Equation simplification is about reducing an equation to its simplest form to make it easy to solve. In our exercise, we started with a more complex equation and simplified it step-by-step.
To simplify equations:
  • Combine like terms, where possible. In this problem, we combined \(x + 2x\) to give \(3x\).
  • Operate on both sides of the equation to isolate the variable. Here, subtracting 2 from both sides helped us simplify to \(3x = 15\).
This strategy helps streamline the problem-solving process and brings you closer to finding the solution.
Problem Verification
Verifying your solution ensures that you have the correct answer. Even when equations suggest a certain result, it's essential to double-check.
Problem verification involves:
  • Substituting the solution back into the original equation.
  • Checking if the equation holds true with the obtained values.
In our example, the solution was verified by checking that the sum of both string pieces equaled 17 feet: \(5 + 12 = 17\). This confirms correctness and gives confidence that all steps were followed accurately.

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