In Exercises 1–26, solve and check each linear equation. 2 - (7x + 5) = 13 - 3x

Verified step by step guidance

1. Start by rewriting the equation to clearly see all terms: \$2 - (7x + 5) = 13 - 3x$.
2. Distribute the negative sign across the parentheses on the left side: \$2 - 7x - 5 = 13 - 3x$.
3. Combine like terms on the left side: \((2 - 5) - 7x = 13 - 3x\), which simplifies to \(-3 - 7x = 13 - 3x\).
4. Get all variable terms on one side and constants on the other by adding \$7x\( to both sides and adding \)3\( to both sides: \)-3 - 7x + 7x + 3 = 13 - 3x + 7x + 3\( which simplifies to \)0 = 16 + 4x$.
5. Isolate \(x\) by subtracting \$16\( from both sides and then dividing both sides by \)4\(: \)-16 = 4x\( then \)x = \frac{-16}{4}$.

Key Concepts

Linear Equations

A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. It forms a straight line when graphed and typically has the form ax + b = c. Solving linear equations involves finding the value of the variable that makes the equation true.

Distributive Property

The distributive property states that a(b + c) = ab + ac. It is used to eliminate parentheses by multiplying the term outside the parentheses by each term inside. Applying this property simplifies expressions and is essential for solving equations like 2 - (7x + 5) = 13 - 3x.

Checking Solutions

After solving an equation, substituting the solution back into the original equation verifies its correctness. This step ensures no mistakes were made during manipulation and confirms the solution satisfies the equation.