In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 10x + 3 = 8x + 3

Verified step by step guidance

1. Start by simplifying the equation: 10x + 3 = 8x + 3.
2. Subtract 8x from both sides to isolate the variable terms on one side: 10x - 8x + 3 = 3.
3. Simplify the left side: 2x + 3 = 3.
4. Subtract 3 from both sides to isolate the term with the variable: 2x = 0.
5. Divide both sides by 2 to solve for x: x = 0.

Key Concepts

Types of Equations

In algebra, 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 determining the nature of the solution set for any given equation.

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. This typically includes combining like terms, using inverse operations, and simplifying both sides of the equation. In the given equation, manipulating the terms will reveal whether the equation holds true for all values or just specific ones.

Checking Solutions

After solving an equation, it is essential to check the solution by substituting it back into the original equation. This verification process confirms whether the solution is valid and helps identify the type of equation. For instance, if both sides of the equation are equal after substitution, it indicates the nature of the equation as either an identity or a conditional equation.