In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 5x + 9 = 9(x + 1) - 4x

Verified step by step guidance

1. Distribute the 9 on the right side of the equation: 9(x + 1) becomes 9x + 9.
2. Rewrite the equation: 5x + 9 = 9x + 9 - 4x.
3. Combine like terms on the right side: 9x - 4x becomes 5x, so the equation is 5x + 9 = 5x + 9.
4. Subtract 5x from both sides of the equation to isolate the constant terms: 9 = 9.
5. Since the resulting statement is true for all values of x, determine that the equation is an identity.

Key Concepts

Solving Linear Equations

Solving linear equations involves finding the value of the variable that makes the equation true. This typically requires isolating the variable on one side of the equation through operations such as addition, subtraction, multiplication, and division. In the given equation, simplifying both sides will help identify the solution.

Types of Equations

Equations can be classified into three types: identities, conditional equations, and inconsistent equations. An identity holds true for all values of the variable (e.g., 0 = 0), a conditional equation is true for specific values (e.g., x = 2), and an inconsistent equation has no solution (e.g., 0 = 5). Understanding these classifications is crucial for interpreting the results after solving the equation.

Simplifying Expressions

Simplifying expressions involves combining like terms and applying the distributive property to make equations easier to solve. In the context of the given equation, distributing the 9 on the right side and combining terms will lead to a clearer path to finding the solution. This step is essential for accurately determining the nature of the equation.