1. **Stating the problem:** We are given the objective function $$Z = 18x + 10y$$ with a value of $$Z = 134$$ and several points and constraints: - Points: $$A_1(5,0), A_2(15,0), B_1(0,20), B_2(0,10), P(3,8)$$ - Constraints: - $$4x + y = 20$$ - $$18x + 10y = 134$$ - $$2x + 3y = 30$$ We want to analyze the function and constraints, and verify the value of $$Z$$ at given points. 2. **Formula and rules:** The objective function is linear: $$Z = 18x + 10y$$. To find $$Z$$ at any point $$(x,y)$$, substitute the coordinates into the formula. 3. **Evaluate $$Z$$ at given points:** - At $$A_1(5,0)$$: $$Z = 18(5) + 10(0) = 90 + 0 = 90$$ - At $$A_2(15,0)$$: $$Z = 18(15) + 10(0) = 270 + 0 = 270$$ - At $$B_1(0,20)$$: $$Z = 18(0) + 10(20) = 0 + 200 = 200$$ - At $$B_2(0,10)$$: $$Z = 18(0) + 10(10) = 0 + 100 = 100$$ - At $$P(3,8)$$: $$Z = 18(3) + 10(8) = 54 + 80 = 134$$ 4. **Check constraints for point $$P(3,8)$$:** - $$4x + y = 4(3) + 8 = 12 + 8 = 20$$ (satisfies first constraint) - $$18x + 10y = 18(3) + 10(8) = 54 + 80 = 134$$ (matches given $$Z$$) - $$2x + 3y = 2(3) + 3(8) = 6 + 24 = 30$$ (satisfies third constraint) 5. **Interpretation:** Point $$P(3,8)$$ satisfies all constraints and yields $$Z = 134$$, which matches the given value. **Final answer:** The value of $$Z$$ at point $$P(3,8)$$ is $$\boxed{134}$$, and this point satisfies all given constraints.