In Exercises 15–35, solve each equation. Then state whether the equation is an identity, a conditional equation, or an inconsistent equation. 3-5(2x + 1) - 2(x-4) = 0

Verified step by step guidance

1. Distribute the -5 across the terms inside the parentheses: \(-5(2x + 1)\) becomes \(-10x - 5\).
2. Distribute the -2 across the terms inside the parentheses: \(-2(x - 4)\) becomes \(-2x + 8\).
3. Substitute the distributed terms back into the equation: \(3 - 10x - 5 - 2x + 8 = 0\).
4. Combine like terms: \(3 - 5 + 8\) and \(-10x - 2x\) to simplify the equation.
5. Rearrange the equation to isolate the variable \(x\) and determine if the equation is an identity, conditional, or inconsistent.

Key Concepts

Solving Linear Equations

Solving linear equations involves isolating the variable to find its value. This typically requires applying algebraic operations such as addition, subtraction, multiplication, and division to both sides of the equation. In the given equation, simplifying and combining like terms will lead to a solution for 'x'.

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 helps in interpreting the results after solving the equation.

Combining Like Terms

Combining like terms is a fundamental algebraic technique used to simplify expressions. It involves adding or subtracting coefficients of terms that have the same variable raised to the same power. In the context of the given equation, this step is crucial for reducing the equation to a simpler form, making it easier to solve for 'x'.