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. Distribute the 4 on the left side: 4(x + 5) becomes 4x + 20.
2. Rewrite the equation: 4x + 20 = 21 + 4x.
3. Subtract 4x from both sides to eliminate the variable on one side: 4x + 20 - 4x = 21 + 4x - 4x.
4. Simplify both sides: 20 = 21.
5. Since 20 does not equal 21, the equation is inconsistent.

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 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, a conditional equation is true for specific values, and an inconsistent equation has no solution. Understanding these classifications is essential for determining the nature of the equation after solving it.

Verification of Solutions

Once a solution is found, it is important to verify it by substituting the value back into the original equation. This step confirms whether the solution satisfies the equation. Additionally, this verification process aids in determining the type of equation, as it reveals if the equation holds true universally, conditionally, or not at all.