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When solving algebraic equations, it is essential to verify if a proposed solution is correct. Solution verification involves substituting the solution back into the original equation to see if it satisfies the equation. This means you compare the left-hand side and the right-hand side of the equation after substitution.
Here’s why it’s important: - It confirms that the solution is correct, providing confidence in your answer. - It helps detect any potential errors made during solving or simplification. To verify if 18 is a solution for our equation, after simplifying and substituting 18 for \(x\), we calculate: \[16 = 18 - 2\] Both sides match as 16, confirming that 18 is indeed the solution. Verification serves as a critical final step that ensures accuracy.

Linear equations are equations of the first degree, meaning they involve variables raised to the power of one. A linear equation typically takes the form \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.

They are foundational in algebra because: - They establish relationships that can portray real-world situations, like calculating distances, costs, etc. - They are often the stepping stone for more complex equations encountered in higher math. In the original exercise, the equation \(16 = 2(x-1)-x\) is a linear equation. Solving it involved balancing both sides to find \(x\). The goal is to isolate the variable on one side to evaluate its value using basic operations like addition and subtraction.

Equation simplification refers to reducing an equation to its simplest form, making it easier to solve. This involves applying mathematical operations to eliminate parentheses, combine like terms, and simplify expressions. Simplification is crucial because: - It makes the equation more understandable and visually clearer. - It aids in efficiently finding the variable's value.

For the given equation \(16 = 2(x-1) - x\): - We first distribute the 2 in \(2(x-1)\) to get \(2x - 2\). - Then, by subtracting \(x\) from \(2x\), the equation becomes \(16 = x - 2\). Through this process, the equation is simplified from a complex expression to a direct one that clearly shows the relationship and makes finding \(x\) easier.