In Exercises 1–26, solve and check each linear equation. 16 = 3(x - 1) - (x - 7)

Verified step by step guidance

1. Start by expanding the expressions on the right side of the equation: apply the distributive property to both terms. For \(3(x - 1)\), multiply 3 by each term inside the parentheses, and for \(-(x - 7)\), distribute the negative sign to both terms inside the parentheses. This gives you \(3 \times x - 3 \times 1 - 1 \times x + 1 \times 7\).
2. Simplify the right side by performing the multiplications and combining like terms. This will result in an expression in terms of \(x\) and constants.
3. Rewrite the equation with the simplified right side, so it looks like \(16 = \text{(simplified expression)}\).
4. Isolate the variable term by moving all terms involving \(x\) to one side of the equation and constants to the other side. You can do this by adding or subtracting terms on both sides.
5. Solve for \(x\) by dividing both sides of the equation by the coefficient of \(x\). After finding \(x\), 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 3(x - 1) by multiplying 3 with both x 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 after distribution in 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 on one side using inverse operations like addition, subtraction, multiplication, or division.