Three spices have unknown heat ratings. Taking the average of two of them and adding the third gives 35, 27 and 25 for the three possible choices. Find the largest of the three ratings.

Step-by-step solution

1. 1

   Clear the halves before anything else. Multiplying each equation by $2$ turns the system into integer coefficients, which makes the symmetry visible:

   $x+y+2z=70,\qquad x+2y+z=54,\qquad 2x+y+z=50.$

2. 2

   Exploit the symmetry: add all three equations. Each variable appears once with coefficient $2$ and twice with coefficient $1$, so every variable picks up a total coefficient of $4$:

   $4x+4y+4z=70+54+50=174\quad\Longrightarrow\quad x+y+z=\frac{174}{4}=43.5.$

   This single number is the key — it turns each of the three equations into a one-line subtraction.

3. 3

   Subtract the total from each doubled equation. In $x+y+2z=70$ the left side is exactly $(x+y+z)+z$, so

   $z=70-43.5=26.5.$

   The same trick on the other two gives

   $y=54-43.5=10.5,\qquad x=50-43.5=6.5.$

4. 4

   Verify all three original conditions. With $x=6.5$, $y=10.5$, $z=26.5$:

   $\frac{6.5+10.5}{2}+26.5=8.5+26.5=35,\qquad \frac{6.5+26.5}{2}+10.5=16.5+10.5=27,$

   $\frac{10.5+26.5}{2}+6.5=18.5+6.5=25.$

   All three match.

5. 5

   Pick out the largest rating. Comparing $6.5$, $10.5$ and $26.5$, the hottest spice has rating

   $z=26.5=\tfrac{53}{2}.$

   Note the shortcut the structure hands you: the largest rating always pairs with the largest of the three given totals, because $z=(\text{its equation's doubled value})-43.5$.