In Exercises 71–78, solve each equation. Then determine whether the equation is an identity, a conditional equation, or an inconsistent equation. 4x + 7 = 7(x + 1) - 3x

Verified step by step guidance

1. Start by simplifying both sides of the equation. On the right-hand side, distribute the 7 across the terms inside the parentheses: 7(x + 1) becomes 7x + 7. The equation now looks like 4x + 7 = 7x + 7 - 3x.
2. Combine like terms on the right-hand side. Combine 7x and -3x to get 4x. The equation now simplifies to 4x + 7 = 4x + 7.
3. Next, subtract 4x from both sides of the equation to isolate the constants. This simplifies the equation to 7 = 7.
4. Analyze the resulting equation. Since 7 = 7 is always true, the original equation is true for all values of x.
5. Conclude that the equation is an identity because it holds true for all values of x.

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, 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 solution 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 7 on the right side and combining terms will lead to a clearer path to finding the solution. This step is crucial for accurately determining the type of equation.