If $2p + 3q = 18$ and $2p - q = 2$, then $2p + q$ is

### Concept & Strategy This is an **Algebraic Pattern Recognition** problem masked as a standard simultaneous equations question. While you can solve for $p$ and $q$ individually using standard elimination or substitution, a highly optimized strategy involves manipulating the entire equations to instantly produce the target expression ($2p + q$). ### Step-by-Step Solution (Standard Method) * **Step 1: Set up the equations** Eq 1: $2p + 3q = 18$ Eq 2: $2p - q = 2$ * **Step 2: Eliminate $p$ to find $q$** Since both equations have a $2p$ term, subtract Eq 2 from Eq 1: $$(2p + 3q) - (2p - q) = 18 - 2$$ $$2p + 3q - 2p + q = 16$$ $$4q = 16$$ $$q = 4$$ * **Step 3: Substitute to find $2p$** Substitute $q = 4$ into Eq 2: $$2p - 4 = 2$$ $$2p = 6$$ *(Note: No need to find p=3, as our target expression needs 2p)* * **Step 4: Calculate the target expression** The question asks for $2p + q$: $$6 + 4 = 10$$ ### Exam Strategy & Shortcut **The Equation Addition Trick:** Always check if adding or subtracting the given equations directly creates the target expression. Let's add the two equations together: Eq 1 + Eq 2: $(2p + 3q) + (2p - q) = 18 + 2$ This simplifies to: $4p + 2q = 20$ Now, look at what we need: $2p + q$. Simply divide the combined equation by 2: $$\frac{4p + 2q}{2} = \frac{20}{2}$$ $$2p + q = 10$$ You arrived at the answer in one step, completely bypassing the need to solve for individual variables! ### Common Pitfall The biggest mistake is wasting time. Many students will systematically solve for $q=4$, then $p=3$, and then substitute both back into $2p + q = 2(3) + 4 = 10$. While mathematically correct, this brute-force approach eats up precious seconds in a competitive exam where shortcuts like equation addition are designed to be used. ### Final Answer Therefore, the correct answer is **10**.