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

Verified step by step guidance

1. Distribute the 2 on the left side: \(2(x - 1)\) becomes \(2x - 2\).
2. Distribute the -3 on the right side: \(-3(x + 1)\) becomes \(-3x - 3\).
3. Rewrite the equation with the distributed terms: \(2x - 2 + 3 = x - 3x - 3\).
4. Combine like terms on both sides: On the left, \(-2 + 3\) becomes \(1\), so \(2x + 1\). On the right, \(x - 3x\) becomes \(-2x\), so \(-2x - 3\).
5. Add \(2x\) to both sides to get all \(x\) terms on one side: \(2x + 2x + 1 = -2x + 2x - 3\).

Key Concepts

Linear Equations

A linear equation is an algebraic expression that represents a straight line when graphed. It typically takes the form ax + b = c, where a, b, and c are constants. Solving a linear equation involves finding the value of the variable that makes the equation true, which often requires isolating the variable on one side of the equation.

Distributive Property

The distributive property is a fundamental algebraic principle that states a(b + c) = ab + ac. This property allows you to multiply a single term by two or more terms inside parentheses. In the context of solving linear equations, applying the distributive property is essential for simplifying expressions and combining like terms effectively.

Checking Solutions

Checking solutions involves substituting the found value of the variable back into the original equation to verify its correctness. This step ensures that the solution satisfies the equation and helps identify any potential errors made during the solving process. It is a crucial practice in algebra to confirm that the solution is valid.