Textbook Question
Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. (3x+1)/3 - 13/2 = (1-x)/4
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Ch. 1 - Equations and Inequalities
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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Table of contents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Chapter 2, Problem 28
The length of a rectangular pool is 6 meters less than twice the width. If the pool's perimeter is 126 meters, what are its dimensions?
Verified step by step guidance1
Define variables: Let the width of the pool be \(w\) meters. Then, the length of the pool is \(2w - 6\) meters, since it is 6 meters less than twice the width.
Recall the formula for the perimeter of a rectangle: \(P = 2 \times (\text{length} + \text{width})\). Here, the perimeter \(P\) is given as 126 meters.
Set up the equation using the perimeter formula: \(126 = 2 \times ((2w - 6) + w)\).
Simplify the equation inside the parentheses: \(126 = 2 \times (3w - 6)\), then distribute the 2: \(126 = 6w - 12\).
Solve the linear equation for \(w\): add 12 to both sides to get \(126 + 12 = 6w\), then divide both sides by 6 to find \(w\). Once \(w\) is found, substitute back to find the length \(2w - 6\).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Formulating Algebraic Expressions from Word Problems
This involves translating verbal descriptions into mathematical expressions or equations. For example, 'length is 6 meters less than twice the width' can be written as L = 2W - 6. This step is crucial for setting up equations that represent the problem.
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Introduction to Algebraic Expressions
Perimeter of a Rectangle
The perimeter of a rectangle is the total distance around it, calculated as P = 2(length + width). Understanding this formula allows you to relate the given perimeter to the dimensions of the pool and form an equation to solve.
Solving Linear Equations
Once the problem is expressed as an equation, solving linear equations involves isolating the variable to find its value. This includes combining like terms, using inverse operations, and checking solutions for accuracy.
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Related Practice
Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = x3 - 1
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Textbook Question
Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = x3
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Textbook Question
In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 3x - 7 ≥ 13
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Textbook Question
Exercises 27–40 contain linear equations with constants in denominators. Solve each equation. 20 - x/3 = x/2
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Textbook Question
In all exercises, other than exercises with no solution, use interval notation to express solution sets and graph each solution set on a number line. In Exercises 27–50, solve each linear inequality. 5x + 11 < 26
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