Solve each linear equation. See Examples 1–3.
(3x - 1)/4 + (x + 3)/6 = 3

Verified step by step guidance

1. Identify the given equation: \(\frac{3x - 1}{4} + \frac{x + 3}{6} = 3\).
2. Find the least common denominator (LCD) of the fractions, which is the least common multiple of 4 and 6. The LCD is 12.
3. Multiply every term in the equation by the LCD (12) to eliminate the denominators: \(12 \times \frac{3x - 1}{4} + 12 \times \frac{x + 3}{6} = 12 \times 3\).
4. Simplify each term after multiplication: \(12 \times \frac{3x - 1}{4} = 3(3x - 1)\) and \(12 \times \frac{x + 3}{6} = 2(x + 3)\), so the equation becomes \$3(3x - 1) + 2(x + 3) = 36$.
5. Expand the parentheses and combine like terms: \$9x - 3 + 2x + 6 = 36\(, then simplify to \)11x + 3 = 36\(. From here, isolate \)x$ by subtracting 3 from both sides and then dividing by 11.

Key Concepts

Solving 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. Solving such equations involves isolating the variable on one side to find its value. Techniques include combining like terms, using inverse operations, and maintaining equality by performing the same operation on both sides.

Working with Fractions in Equations

When equations contain fractions, it is important to find a common denominator or multiply through by the least common denominator (LCD) to eliminate fractions. This simplifies the equation and makes it easier to solve. Care must be taken to apply operations uniformly to both sides to preserve equality.

Combining Like Terms and Simplification

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. Simplification reduces the equation to its simplest form, making it easier to isolate the variable. This step is crucial before solving for the unknown to avoid errors and confusion.