Let's analyze the algebraic word problems from the image:

1. First Problem: "Product of two numbers is 120 and their sum is 26. Find the numbers."

   • We can let the numbers be $x$ and $y$. The problem provides two equations: $xy = 120$ $x + y = 26$

2. Second Problem: "Two different numbers differ by 2, and the square of the smaller number is equal to the product of the two numbers. Find the numbers."

   • Let the numbers be $x$ and $y$ where $x < y$. The conditions give: $y - x = 2$ $x^2 = xy$

3. Third Problem: "The perimeter of a rectangular field is 80 meters, and its area is 384 square meters. Find the dimensions of the field."

   • Let the length and width be $l$ and $w$, respectively. The conditions are: $2(l + w) = 80$ $lw = 384$

4. Fourth Problem: "The sum of three consecutive integers is 60. Find the integers."

   • Let the integers be $x-1$, $x$, and $x+1$. The equation is: $(x - 1) + x + (x + 1) = 60$

5. Fifth Problem: "The sum of two numbers is 19, and their product is 84. Find the numbers."

   • Let the numbers be $x$ and $y$. The equations are: $x + y = 19$ $xy = 84$

6. Sixth Problem: "Two angles are supplementary, and the larger is 36 degrees more than the smaller. Find the angles."

   • Let the smaller angle be $x$ and the larger angle be $y$. The conditions are: $x + y = 180^\circ$ $y = x + 36^\circ$

7. Seventh Problem: "A bus travels 60 km in the same time that a train travels 84 km. If the speed of the train is 20 km/h more than that of the bus, find the speed of each."

   • Let the speed of the bus be $x$ km/h and the train be $y$ km/h. The conditions are: $y = x + 20$ $\frac{60}{x} = \frac{84}{y}$

8. Eighth Problem: "Show that the sum of three consecutive even numbers and three consecutive odd numbers is always 3 times the middle number of the odd numbers."

   • Let the even numbers be $2n, 2n+2, 2n+4$ and the odd numbers be $2m+1, 2m+3, 2m+5$. The expression to prove is: $(2n + 2n+2 + 2n+4) + (2m+1 + 2m+3 + 2m+5) = 3(2m+3)$

Would you like a detailed solution for any of these problems?

Here are some questions related to the information:

1. How do you solve a system of equations to find two numbers?
2. What is the difference between supplementary and complementary angles?
3. How can we derive the formula for the perimeter of a rectangle?
4. What does it mean for three numbers to be consecutive integers?
5. How can the quadratic formula be applied to find two numbers whose product and sum are known?
6. What are the properties of even and odd numbers?
7. How do we solve a word problem involving speeds and distances?
8. What is the relationship between the arithmetic mean and the middle number in a sequence?

Tip: When solving word problems, always define your variables clearly and write down the equations that correspond to the conditions given in the problem.