1. **Problem Statement:** Find the individual concentrations of ingredients A, B, and C given the system: $$\begin{cases} a + b + c = 120 \\ 2a + 3b + c = 250 \\ a + 4b + 2c = 310 \end{cases}$$ 2. **Formula and Rules:** We solve this system of linear equations using substitution or elimination. The goal is to find values of $a$, $b$, and $c$ that satisfy all three equations simultaneously. 3. **Step 1: Express $c$ from the first equation:** $$c = 120 - a - b$$ 4. **Step 2: Substitute $c$ into the second and third equations:** Second equation: $$2a + 3b + (120 - a - b) = 250$$ Simplify: $$2a + 3b + 120 - a - b = 250$$ $$a + 2b + 120 = 250$$ $$a + 2b = 130$$ Third equation: $$a + 4b + 2(120 - a - b) = 310$$ Simplify: $$a + 4b + 240 - 2a - 2b = 310$$ $$-a + 2b + 240 = 310$$ $$-a + 2b = 70$$ 5. **Step 3: Solve the two-variable system:** $$\begin{cases} a + 2b = 130 \\ -a + 2b = 70 \end{cases}$$ Add the two equations: $$(a - a) + (2b + 2b) = 130 + 70$$ $$4b = 200$$ $$b = 50$$ 6. **Step 4: Find $a$ using $a + 2b = 130$:** $$a + 2(50) = 130$$ $$a + 100 = 130$$ $$a = 30$$ 7. **Step 5: Find $c$ using $c = 120 - a - b$:** $$c = 120 - 30 - 50 = 40$$ **Final answer:** $$a = 30, \quad b = 50, \quad c = 40$$