Determine whether each equation is an identity, a conditional equation, or a contradiction. Give the solution set. -6(2x+1) - 3(x-4) = -15x+1

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Start by expanding both sides of the equation. Use the distributive property to remove the parentheses: expand \(-6(2x+1)\) and \(-3(x-4)\) on the left side.

After expanding, combine like terms on the left side to simplify the expression into the form \(ax + b\).

Rewrite the equation with the simplified left side and the right side \(-15x + 1\).

Next, get all terms involving \(x\) on one side and constant terms on the other side by adding or subtracting terms accordingly.

Finally, analyze the resulting equation: if it simplifies to a true statement for all \(x\), it is an identity; if it is true for specific \(x\) values, it is a conditional equation with those solutions; if it results in a false statement, it is a contradiction with an empty solution set.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Types of Equations: Identity, Conditional, and Contradiction

An identity is true for all values of the variable, a conditional equation is true for specific values, and a contradiction has no solution. Identifying the type helps determine the solution set and the nature of the equation.

Solving Linear Equations

Solving linear equations involves simplifying both sides, combining like terms, and isolating the variable. This process helps find the values of the variable that satisfy the equation or determine if no or all values work.

Distributive Property

The distributive property allows multiplication over addition or subtraction, such as a(b + c) = ab + ac. Applying this property correctly is essential to simplify expressions and solve equations accurately.