If $x = 1 - q$ and $y = 2q + 1$, then for what value of $q$, $x$ is equal to $y$?

### Concept & Formula This is a fundamental linear equation problem using substitution. When two distinct variables ($x$ and $y$) are expressed in terms of a third common variable ($q$), setting the two variables equal to each other allows you to solve for the value of that third variable. $$ \text{If } x = f(q) \text{ and } y = g(q), \text{ and } x = y, \text{ then } f(q) = g(q) $$ ### Step-by-Step Solution * **Given:** * Equation 1: $x = 1 - q$ * Equation 2: $y = 2q + 1$ * Condition: $x = y$ * **Calculation:** 1. Set the expressions for $x$ and $y$ equal to each other based on the given condition. 2. $1 - q = 2q + 1$. 3. To isolate $q$, start by adding $q$ to both sides of the equation to group the variables on the right. 4. $1 = 3q + 1$. 5. Subtract $1$ from both sides to isolate the term with $q$. 6. $1 - 1 = 3q$. 7. $0 = 3q$. 8. Divide both sides by $3$. 9. $q = 0$. ### Exam Strategy & Shortcut For simple linear equations with multiple choice options, plugging the options directly into the expressions is often faster than algebra. Try (b) $q = 0$: $x = 1 - 0 = 1$ $y = 2(0) + 1 = 1$ Since $1 = 1$, option (b) is instantly verified as the correct answer. ### Common Pitfall A minor but common pitfall in algebraic manipulation is mishandling the signs when moving terms across the equals sign. For example, moving $-q$ to the right side and accidentally subtracting it again (yielding $q$) instead of adding it (yielding $3q$). Carefully double-check sign changes during grouping. ### Final Answer Therefore, the correct answer is 0.