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
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 26
A rectangular swimming pool is three times as long as it is wide. If the perimeter of the pool is 320 feet, what are its dimensions?
Verified step by step guidance1
Let the width of the rectangular pool be represented by \(w\). Since the pool is three times as long as it is wide, express the length as \$3w$.
Recall the formula for the perimeter \(P\) of a rectangle: \(P = 2 \times (\text{length} + \text{width})\). Substitute the expressions for length and width into this formula: \(320 = 2 \times (3w + w)\).
Simplify the expression inside the parentheses: \(3w + w = 4w\), so the equation becomes \(320 = 2 \times 4w\).
Simplify the right side: \(2 \times 4w = 8w\), so the equation is \(320 = 8w\). Solve for \(w\) by dividing both sides by 8.
Once you find \(w\), calculate the length by multiplying \(w\) by 3 to get \$3w$. These values represent the width and length of the pool, respectively.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Perimeter of a Rectangle
The perimeter of a rectangle is the total distance around it, calculated by adding twice the length and twice the width (P = 2L + 2W). Understanding this formula is essential to relate the given perimeter to the pool's dimensions.
Algebraic Representation of Relationships
Translating the problem's conditions into algebraic expressions is crucial. Here, the length is three times the width, so L = 3W. This relationship allows substitution into the perimeter formula to solve for one variable.
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Solving Linear Equations
After forming an equation with one variable, solving linear equations involves isolating the variable to find its value. This step is necessary to determine the pool's width and then calculate its length.
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