Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

Linear Equations

College Algebra

1. Equations & Inequalities

Linear Equations

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1:14 minutes

Problem 15

Textbook Question

In Exercises 15–35, 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 guidance

Start by isolating the variable term on one side of the equation. To do this, add 5 to both sides of the equation: \(2x - 5 + 5 = 7 + 5\).

Simplify both sides of the equation: \(2x = 12\).

Next, solve for \(x\) by dividing both sides of the equation by 2: \(\frac{2x}{2} = \frac{12}{2}\).

Simplify the division to find the value of \(x\): \(x = 6\).

Since the equation has a specific solution \(x = 6\), it is a conditional equation.

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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 finding the value of the variable that makes the equation true. In the equation 2x - 5 = 7, we isolate x by performing inverse operations, such as adding 5 to both sides and then dividing by 2. This process is fundamental in algebra as it allows us to determine specific values that satisfy the equation.

Types of Equations

Equations can be classified into three main types: identities, conditional equations, and inconsistent equations. An identity is true for all values of the variable, a conditional equation is true for specific values, and an inconsistent equation has no solutions. Understanding these classifications helps in analyzing the nature of the solutions obtained from solving an equation.

Verification of Solutions

After solving an equation, it is essential to verify the solution by substituting it back into the original equation. This step confirms whether the derived value satisfies the equation. For example, substituting the solution of 2x - 5 = 7 back into the equation allows us to check if both sides are equal, thus validating the solution and determining the type of equation.