Textbook Question
In Exercises 1–26, solve and check each linear equation. 3(x - 8) = x
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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 15
Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2x-5 = 7
Verified step by step guidance1
Start with the given equation: \(2x - 5 = 7\).
Add 5 to both sides of the equation to isolate the term with \(x\): \(2x - 5 + 5 = 7 + 5\), which simplifies to \(2x = 12\).
Divide both sides of the equation by 2 to solve for \(x\): \(\frac{2x}{2} = \frac{12}{2}\), which simplifies to \(x = 6\).
Interpret the solution: since \(x = 6\) satisfies the original equation, this is a conditional equation with one solution.
Conclude that the equation is conditional because it is true for a specific value of \(x\) (namely \(x = 6\)), not for all values (identity) or no values (inconsistent).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Solving Linear Equations
Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, or division. For example, in 2x - 5 = 7, you add 5 to both sides and then divide by 2 to find x.
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Types of Equations: Identity, Conditional, and Inconsistent
An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solution. After solving, you determine which type applies based on the solution set.
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Checking Solutions
After finding a solution, substitute it back into the original equation to verify its validity. This step confirms whether the solution satisfies the equation, helping to classify the equation correctly.
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Related Practice
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Solve each equation in Exercises 15–34 by the square root property.
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In Exercises 15–26, use graphs to find each set. (- 3, 0) ∩ [- 1, 2]
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After a 20% reduction, you purchase a television for \$336. What was the television's price before the reduction?
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Graph each equation in Exercises 13 - 28. Let x = - 3, - 2, - 1, 0, 1, 2, 3
y = x - 2
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