If $2x + y = 15$, $2y + z = 25$ and $2z + x = 26$, what is the value of $z$?

### Concept & Formula This is a classic "Cyclic" system of equations where the variables shift symmetrically across the equations. The fastest way to break a cyclic system is to add all the equations together first, revealing a master equation of $(x + y + z)$. ### Step-by-Step Solution **Given:** 1. $2x + y = 15$ 2. $2y + z = 25$ 3. $x + 2z = 26$ **Calculation:** * **Step 1: Exploit the cyclic symmetry** Add all three equations together vertically: $(2x + x) + (y + 2y) + (z + 2z) = 15 + 25 + 26$ $3x + 3y + 3z = 66$ Divide the entire equation by 3: $x + y + z = 22$ --- (Master Equation) * **Step 2: Isolate the target variables** We need $z$. Let's find a way to substitute $x+y$ out. From Equation 1, we know $y = 15 - 2x$. While we *could* use substitution, there's an easier route via algebraic manipulation. Let's manipulate the given equations to find $x$: From (1): $y = 15 - 2x$ From (2): $z = 25 - 2y = 25 - 2(15 - 2x) = -5 + 4x$ From (3): Substitute $z$ into $x + 2z = 26$: $x + 2(-5 + 4x) = 26$ $x - 10 + 8x = 26$ $9x = 36 \implies x = 4$ * **Step 3: Solve for $z$** Now plug $x = 4$ into our expression for $z$: $z = -5 + 4(4) = -5 + 16 = 11$ ### Exam Strategy & Shortcut Whenever you see a cyclic pattern like $ax+y, ay+z, az+x$, adding them all to find $x+y+z$ is usually the intended trick. While we pivoted to substitution to find the exact value here, having the master equation $x+y+z=22$ provides a fantastic 5-second verification tool at the end. (Check: $x=4$, so $y = 15-8=7$, so $z = 22-4-7 = 11$. Correct!). ### Common Pitfall A common time-waster is trying to use standard two-equation elimination on cyclic systems. Because every equation lacks one variable entirely, trying to stack and subtract them directly leads to messy, frustrating fractions. Always add them first. ### Final Answer **Therefore, the correct answer is 11.**