Solve each linear equation. See Examples 1–3. 6(3x - 1) = 8 - (10x - 14)

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, multiply 6 by each term inside the parentheses: \(6(3x - 1) = 6 \times 3x - 6 \times 1\). For the right side, distribute the negative sign across the terms inside the parentheses: \(-(10x - 14) = -10x + 14\).
2. Rewrite the equation after distribution: the left side becomes \$18x - 6\(, and the right side becomes \)8 - 10x + 14$. Combine like terms on the right side by adding the constants \(8\) and \(14\).
3. Next, collect all variable terms on one side and constants on the other. You can add or subtract terms from both sides to isolate \(x\). For example, add \$10x$ to both sides and add \(6\) to both sides to move terms accordingly.
4. After rearranging, combine like terms on each side to simplify the equation to the form \(ax = b\), where \(a\) and \(b\) are constants.
5. Finally, solve for \(x\) by dividing both sides of the equation by the coefficient \(a\). This will give you the solution for \(x\).

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 6(3x - 1) by distributing 6 to both 3x and -1.

Combining Like Terms

Combining like terms involves adding or subtracting terms that have the same variable raised to the same power. This simplifies expressions and makes solving equations easier, such as combining terms with x on one side of the equation.

Solving Linear Equations

Solving linear equations means finding the value of the variable that makes the equation true. This involves isolating the variable by performing inverse operations like addition, subtraction, multiplication, or division on both sides of the equation.