If $a + b = 5$ and $3a + 2b = 20$, then $(3a + b)$ will be:

### Concept & Strategy This problem requires solving **Simultaneous Linear Equations**. We can use the **Elimination Method** to quickly find the values of $a$ and $b$, and then substitute them into the target expression $(3a + b)$. ### Step-by-Step Solution * **Step 1: Set up the equations** Equation 1: $a + b = 5$ Equation 2: $3a + 2b = 20$ * **Step 2: Equate coefficients for elimination** Let's eliminate $b$. Multiply Equation 1 by 2 so the coefficients of $b$ match: $$2 \times (a + b) = 2 \times 5$$ $$2a + 2b = 10$$ (This is our new Equation 3) * **Step 3: Subtract to find $a$** Subtract Equation 3 from Equation 2: $$(3a + 2b) - (2a + 2b) = 20 - 10$$ $$3a - 2a + 2b - 2b = 10$$ $$a = 10$$ * **Step 4: Find $b$** Substitute $a = 10$ back into Equation 1: $$10 + b = 5$$ $$b = 5 - 10$$ $$b = -5$$ * **Step 5: Evaluate the target expression** The question asks for the value of $3a + b$: $$3(10) + (-5)$$ $$30 - 5 = 25$$ ### Exam Strategy & Shortcut Look for algebraic manipulations that directly yield the target expression without finding individual variables. Notice that we want $(3a + b)$. We are given $(3a + 2b) = 20$. If we subtract $b$ from $20$, we get our answer. How do we get $b$? We know $(a + b) = 5$, which implies $2a + 2b = 10$. Subtracting this from $3a + 2b = 20$ instantly gives $a = 10$. Then $b$ must be $-5$. The target is $3(10) - 5 = 25$. While this still requires solving for the variables, actively looking for these structural relationships builds strong algebraic intuition. ### Common Pitfall The main pitfall is dealing with negative numbers. When substituting $a = 10$ back into $a + b = 5$, students frequently make a sign error, calculating $b = 5$ instead of $b = -5$. This would lead to evaluating $3(10) + 5 = 35$, which is incorrect. Always track your signs carefully when moving terms across the equals sign. ### Final Answer Therefore, the correct answer is **25**.