If $2x + y = 17$; $y + 2z = 15$ and $x + y = 9$, then what is the value of $4x + 3y + z$?

### Concept & Formula When presented with three variables across multiple simple linear equations, look for two equations that share the exact same variables to quickly solve for them. Once you find one variable, a cascading substitution will reveal the rest. ### Step-by-Step Solution **Given:** 1. $2x + y = 17$ 2. $y + 2z = 15$ 3. $x + y = 9$ **Calculation:** * Notice equations (1) and (3) only contain $x$ and $y$. We can subtract equation (3) from equation (1) directly: $(2x + y) - (x + y) = 17 - 9$ $x = 8$ * Substitute $x = 8$ into equation (3) to find $y$: $8 + y = 9 \implies y = 1$ * Substitute $y = 1$ into equation (2) to find $z$: $1 + 2z = 15$ $2z = 14 \implies z = 7$ * Now that we have all values ($x=8$, $y=1$, $z=7$), evaluate the final expression $4x + 3y + z$: $4(8) + 3(1) + 7 = 32 + 3 + 7 = 42$ ### Exam Strategy & Shortcut Scan the equations before putting pen to paper. Spotting that $2x + y$ and $x + y$ can be subtracted to instantly yield $x$ saves you from setting up a messy 3-variable matrix. Always attack the "weakest link"—the equations with the most shared terms. ### Common Pitfall A frequent mistake is blindly trying to add all equations together (a common trick for cyclic equations) without realizing it doesn't yield a clean $x+y+z$ form here. Always assess the structure before applying a generic trick. ### Final Answer **Therefore, the correct answer is None of these.**