In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 2(x-4)+3(x+5)=2x-2

Verified step by step guidance

1. Distribute the 2 into the terms inside the first parenthesis: \(2(x-4) = 2x - 8\).
2. Distribute the 3 into the terms inside the second parenthesis: \(3(x+5) = 3x + 15\).
3. Combine the distributed terms: \(2x - 8 + 3x + 15\).
4. Combine like terms on the left side: \(5x + 7\).
5. Set the equation: \(5x + 7 = 2x - 2\) and solve for \(x\) by isolating \(x\) on one side.

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, you would simplify both sides and combine like terms to find the solution.

Types of Equations

Equations can be classified into three main types: identities, conditional equations, and inconsistent equations. 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. Understanding these classifications helps in determining the nature of the solution after solving the equation.

Combining Like Terms

Combining like terms is a fundamental algebraic technique used to simplify expressions. It involves adding or subtracting terms that have the same variable raised to the same power. In the context of the given equation, this step is crucial for simplifying both sides before isolating the variable, ultimately leading to the solution.