$a$ is greater than $b$ by 2 and $b$ is greater than $c$ by 10. If $a + b + c = 130$, then $(b + c) - a =$ ?

### Concept & Logic To solve a linear equation with three variables, express all variables in terms of a single, central variable. This allows you to combine like terms and solve the equation easily. ### Step-by-Step Solution * **Given:** * $a = b + 2$ * $b = c + 10 \Rightarrow c = b - 10$ * $a + b + c = 130$ * **Calculation:** 1. Substitute the expressions for $a$ and $c$ into the sum equation using $b$ as the base variable: $(b + 2) + b + (b - 10) = 130$ 2. Combine the like terms: $3b - 8 = 130$ 3. Solve for $b$: $3b = 138 \Rightarrow b = 46$ 4. Now find the values of $a$ and $c$: $a = 46 + 2 = 48$ $c = 46 - 10 = 36$ 5. Calculate the final required expression $(b + c) - a$: $(46 + 36) - 48 = 82 - 48 = 34$ ### Exam Strategy & Shortcut You can simplify the final expression algebraically before calculating individual values to save time. We want to find $(b + c) - a$. We know $a = b + 2$ and $c = b - 10$. Substitute these into the target expression: $b + (b - 10) - (b + 2) = b - 12$. Since we found $3b = 138 \Rightarrow b = 46$, the final answer is simply $46 - 12 = 34$. This skips the need to calculate $a$ and $c$ entirely. ### Common Pitfall Students often make sign errors when substituting $(b + 2)$ for $a$ in the final expression, doing $- b + 2$ instead of $-(b + 2)$, which leads to an incorrect final addition. Always use parentheses when substituting algebraic terms. ### Final Answer **Therefore, the correct answer is 34.**