In Exercises 67–70, find all values of x such that y = 0.
y = 2[3x - (4x - 6)] - 5(x - 6)

Verified step by step guidance

1. Start by simplifying the expression for y. Distribute the 2 across the terms inside the brackets: y = 2[3x - (4x - 6)] becomes y = 2[3x - 4x + 6].
2. Simplify the terms inside the brackets: y = 2[-x + 6]. Then distribute the 2 to each term: y = -2x + 12.
3. Now simplify the second part of the equation, -5(x - 6). Distribute the -5: y = -5x + 30.
4. Combine the simplified parts of the equation: y = (-2x + 12) + (-5x + 30). Combine like terms: y = -7x + 42.
5. Set y = 0 to find the values of x: 0 = -7x + 42. Solve for x by isolating x: Subtract 42 from both sides, then divide by -7 to find x.

Key Concepts

Solving Equations

Solving equations involves finding the values of the variable that make the equation true. In this case, we need to determine the values of x for which y equals zero. This typically requires isolating the variable on one side of the equation and simplifying the expression.

Distributive Property

The distributive property states that a(b + c) = ab + ac. This property is essential for simplifying expressions where a term is multiplied by a sum or difference. In the given equation, applying the distributive property will help simplify the expression inside the brackets and the terms outside.

Combining Like Terms

Combining like terms is the process of simplifying an expression by adding or subtracting terms that have the same variable raised to the same power. This step is crucial in solving equations, as it reduces the complexity of the expression, making it easier to isolate the variable and find the solution.