Solve and check each linear equation. 2(x - 1) + 3 = x - 3(x + 1)

Verified step by step guidance

1. Start by expanding both sides of the equation: apply the distributive property to remove parentheses. For the left side, distribute 2 over (x - 1), and for the right side, distribute -3 over (x + 1). This gives you: \$2 \cdot x - 2 \cdot 1 + 3 = x - 3 \cdot x - 3 \cdot 1$.
2. Simplify both sides by performing the multiplications and combining like terms. On the left, simplify \$2x - 2 + 3\(, and on the right, simplify \)x - 3x - 3$.
3. Combine like terms on each side to get a simpler linear equation. For example, combine constants on the left side and combine the \(x\) terms on the right side.
4. Next, get all variable terms on one side and constants on the other side. You can do this by adding or subtracting terms from both sides of the equation to isolate \(x\).
5. Finally, solve for \(x\) by dividing or multiplying both sides by the coefficient of \(x\). After finding the value of \(x\), substitute it back into the original equation to check if both sides are equal, confirming your solution.

Key Concepts

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses. For example, a(b + c) = ab + ac. This is essential for simplifying expressions like 2(x - 1) and -3(x + 1) before solving the equation.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. This simplifies the equation and makes it easier to isolate the variable. For instance, terms with x can be combined to reduce complexity.

Solving Linear Equations

Solving linear equations means finding the value of the variable that makes the equation true. This involves isolating the variable on one side using inverse operations like addition, subtraction, multiplication, or division. Checking the solution ensures accuracy.