Textbook Question
Determine whether each relation defines a function, and give the domain and range. See Examples 1 – 4.
Ch. R - Algebra Review
Lial12th EditionTrigonometryISBN: 9780136552161
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Table of contents
- Ch. R - Algebra Review381
- Ch. 1 - Trigonometric Functions188
- Ch. 2 - Acute Angles and Right Triangles168
- Ch. 3 - Radian Measure and The Unit Circle155
- Ch. 4 - Graphs of the Circular Functions100
- Ch. 5 - Trigonometric Identities238
- Ch. 6 - Inverse Circular Functions and Trigonometric Equations158
- Ch. 7 - Applications of Trigonometry and Vectors148
- Ch. 8 - Complex Numbers, Polar Equations, and Parametric Equations15
Chapter 1, Problem 27
Solve each linear equation. See Examples 1–3.
(3x - 1)/4 + (x + 3)/6 = 3
Verified step by step guidance1
Identify the given equation: \(\frac{3x - 1}{4} + \frac{x + 3}{6} = 3\).
Find the least common denominator (LCD) of the fractions, which is the least common multiple of 4 and 6. The LCD is 12.
Multiply every term in the equation by the LCD (12) to eliminate the denominators: \(12 \times \frac{3x - 1}{4} + 12 \times \frac{x + 3}{6} = 12 \times 3\).
Simplify each term after multiplication: \(12 \times \frac{3x - 1}{4} = 3(3x - 1)\) and \(12 \times \frac{x + 3}{6} = 2(x + 3)\), so the equation becomes \(3(3x - 1) + 2(x + 3) = 36\).
Expand the parentheses and combine like terms: \(9x - 3 + 2x + 6 = 36\), then simplify to \(11x + 3 = 36\). From here, isolate \(x\) by subtracting 3 from both sides and then dividing by 11.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Solving such equations involves isolating the variable on one side to find its value. Techniques include combining like terms, using inverse operations, and maintaining equality by performing the same operation on both sides.
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Solving Linear Equations
Working with Fractions in Equations
When equations contain fractions, it is important to find a common denominator or multiply through by the least common denominator (LCD) to eliminate fractions. This simplifies the equation and makes it easier to solve. Care must be taken to apply operations uniformly to both sides to preserve equality.
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4:02Solving Linear Equations with Fractions
Combining Like Terms and Simplification
Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. Simplification reduces the equation to its simplest form, making it easier to isolate the variable. This step is crucial before solving for the unknown to avoid errors and confusion.
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6:20Example 6
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