Solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 7x + 13 = 2(2x-5) + 3x + 23

Verified step by step guidance

1. Start by expanding the right-hand side of the equation \$7x + 13 = 2(2x - 5) + 3x + 23\(. Use the distributive property to multiply \)2\( by each term inside the parentheses: \)2 \times 2x\( and \)2 \times (-5)$.
2. Rewrite the equation after distribution: \$7x + 13 = 4x - 10 + 3x + 23\(. Next, combine like terms on the right-hand side: combine \)4x\( and \)3x\(, and combine \)-10\( and \)23$.
3. After combining like terms, the equation becomes \$7x + 13 = 7x + 13\(. Now, subtract \)7x$ from both sides to isolate the constants and variables separately.
4. Simplify the resulting equation after subtraction. You should get an equation involving only constants. Analyze this simplified equation to determine if it is always true, sometimes true, or never true.
5. Based on the simplified equation, classify the original equation as an identity (true for all values of \(x\)), a conditional equation (true for specific values of \(x\)), or an inconsistent equation (no solution).

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.

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses, transforming expressions like a(b + c) into ab + ac. This step is essential for simplifying and solving equations involving parentheses.

Types of Equations: Identity, Conditional, and Inconsistent

An identity is true for all variable values, a conditional equation is true for specific values, and an inconsistent equation has no solution. Identifying the type depends on the solution set after solving the equation.