In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4(x + 5) = 21 + 4x

Verified step by step guidance

1. Start by expanding the left side of the equation using the distributive property: \(4(x + 5) = 4 \times x + 4 \times 5\).
2. Rewrite the equation after distribution: \(4x + 20 = 21 + 4x\).
3. Next, subtract \$4x$ from both sides to begin isolating the constants: \(4x + 20 - 4x = 21 + 4x - 4x\).
4. Simplify both sides: \(20 = 21\).
5. Analyze the resulting statement: since \(20 = 21\) is false, conclude whether the equation is an identity, conditional, or inconsistent based on this contradiction.

Key Concepts

Solving Linear Equations

Solving linear equations involves isolating the variable on one side of the equation using inverse operations such as addition, subtraction, multiplication, and division. The goal is to find the value of the variable that makes the equation true.

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. Identifying the type depends on the solution set after simplifying the equation.

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses, e.g., a(b + c) = ab + ac. This property is essential for simplifying expressions before solving equations.