1. **Problem Statement:** During a school fair, tickets for children and adults were sold. Let the number of children tickets be $x$ and adult tickets be $y$. The system of equations is: $$x + y = 120$$ $$2x + 3y = 320$$ Find: a) Number of children tickets sold. b) Number of adult tickets sold. c) Represent the equations graphically. d) State the nature of the solution. e) Calculate total collection if adult ticket price is 30 and child ticket price is 20. 2. **Formula and Rules:** - We solve the system of linear equations using substitution or elimination. - The solution $(x,y)$ satisfies both equations. 3. **Step-by-step Solution:** **Step 1:** From the first equation, express $y$ in terms of $x$: $$y = 120 - x$$ **Step 2:** Substitute $y$ into the second equation: $$2x + 3(120 - x) = 320$$ Simplify: $$2x + 360 - 3x = 320$$ $$\Rightarrow 2x - 3x = 320 - 360$$ $$\Rightarrow -x = -40$$ **Step 3:** Solve for $x$: $$x = 40$$ **Step 4:** Substitute $x=40$ back into $y = 120 - x$: $$y = 120 - 40 = 80$$ **Step 5:** Nature of solution: Since the system has a unique solution $(40, 80)$, the lines intersect at one point. **Step 6:** Total collection: Child tickets price = 20, adult tickets price = 30 Total collection = $20 \times 40 + 30 \times 80$ $$= 800 + 2400 = 3200$$ 4. **Summary:** - Children tickets sold: 40 - Adult tickets sold: 80 - Unique solution (intersecting lines) - Total collection: 3200