Solve: 9(x − 1) = 1 + 3(x−2). (Section 1.4, Example 3)

Verified step by step guidance

1. Distribute the 9 on the left side: 9(x - 1) becomes 9x - 9.
2. Distribute the 3 on the right side: 3(x - 2) becomes 3x - 6.
3. Rewrite the equation with the distributed terms: 9x - 9 = 1 + 3x - 6.
4. Combine like terms on the right side: 1 - 6 becomes -5, so the equation is 9x - 9 = 3x - 5.
5. Subtract 3x from both sides to get all x terms on one side: 9x - 3x - 9 = -5.

Key Concepts

Distributive Property

The Distributive Property states that a(b + c) = ab + ac. This property allows us to multiply a single term by each term inside a set of parentheses. In the context of the equation 9(x - 1), applying the distributive property helps simplify the expression by multiplying 9 with both x and -1.

Combining Like Terms

Combining like terms involves simplifying expressions by adding or subtracting terms that have the same variable raised to the same power. In the equation, after distributing and rearranging, it is essential to combine like terms to isolate the variable and solve for x effectively.

Isolating the Variable

Isolating the variable is a fundamental step in solving equations, where the goal is to get the variable (in this case, x) on one side of the equation by itself. This often involves performing inverse operations, such as addition or subtraction, and division or multiplication, to simplify the equation until x is isolated.