In Exercises 1–26, solve and check each linear equation. 3(x - 8) = x

Verified step by step guidance

1. Start by applying the distributive property to the left side of the equation: multiply 3 by both terms inside the parentheses. This gives you \(3 \times x - 3 \times 8\), which simplifies to \$3x - 24$.
2. Rewrite the equation with the distributed terms: \$3x - 24 = x$.
3. Next, get all the variable terms on one side of the equation. Subtract \(x\) from both sides to isolate the variable terms: \$3x - x - 24 = 0\(, which simplifies to \)2x - 24 = 0$.
4. Now, isolate the variable term by adding 24 to both sides: \$2x - 24 + 24 = 0 + 24\(, simplifying to \)2x = 24$.
5. Finally, solve for \(x\) by dividing both sides by 2: \(\frac{2x}{2} = \frac{24}{2}\), which simplifies to \(x = 12\). After finding this value, substitute it back into the original equation to check if both sides are equal.

Key Concepts

Distributive Property

The distributive property allows you to multiply a single term by each term inside parentheses. For example, in 3(x - 8), you multiply 3 by x and 3 by -8, resulting in 3x - 24. This step simplifies the equation and is essential before combining like terms.

Solving Linear Equations

Solving linear equations involves isolating the variable on one side to find its value. This typically requires combining like terms, using inverse operations such as addition or subtraction, and division or multiplication to simplify the equation to the form x = a number.

Checking Solutions

After finding a solution, substitute it back into the original equation to verify its correctness. This ensures that the solution satisfies the equation and helps identify any errors made during the solving process.