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. Step 1: Begin by simplifying both sides of the equation. Expand the expression on the right-hand side using the distributive property: \( 9(x + 1) \) becomes \( 9x + 9 \). The equation now looks like \( 5x + 9 = 9x + 9 - 4x \).
2. Step 2: Combine like terms on the right-hand side. Combine \( 9x \) and \( -4x \) to get \( 5x \). The equation simplifies to \( 5x + 9 = 5x + 9 \).
3. Step 3: Subtract \( 5x \) from both sides of the equation to isolate the constant terms. This results in \( 9 = 9 \).
4. Step 4: Analyze the simplified equation \( 9 = 9 \). Since this statement is always true, the original equation is an identity. An identity is an equation that is true for all values of the variable.
5. Step 5: Conclude that the equation is an identity and does not depend on the value of \( x \). No further steps are needed.

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, or division. In the given equation, simplifying both sides will help identify the value of 'x'.

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 determining the nature of the solution.

Simplifying Expressions

Simplifying expressions involves combining like terms and performing operations to reduce the equation to its simplest form. This process is essential in solving equations, as it allows for clearer identification of the variable's value. In the provided equation, distributing and combining terms will facilitate the solution process.