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**(A) Price-Supply Equation:**

We are given two points for supply:
- At price $ p = 2.25 $ dollars, the supply is $ x = 7500 $ million bushels.
- At price $ p = 2.36 $ dollars, the supply is $ x = 7900 $ million bushels.

We can use the form $ p = mx + b $ for the price-supply equation, where $ p $ is the price, $ x $ is the supply, $ m $ is the slope, and $ b $ is the y-intercept.

1. **Find the slope** $ m $ between the two points:
   \[
   m = \frac{p_2 - p_1}{x_2 - x_1} = \frac{2.36 - 2.25}{7900 - 7500} = \frac{0.11}{400} = 0.000275
   \]

2. **Substitute one of the points to find $ b $:**
   Using the point $ (7500, 2.25) $:
   \[
   p = mx + b \implies 2.25 = 0.000275(7500) + b
   \]
   \[
   2.25 = 2.0625 + b \implies b = 2.25 - 2.0625 = 0.1875
   \]

Thus, the price-supply equation is:
\[
p = 0.000275x + 0.1875
\]

---

**(B) Price-Demand Equation:**

Similarly, for demand, we are given two points:
- At price $ p = 2.25 $ dollars, the demand is $ x = 7700 $ million bushels.
- At price $ p = 2.36 $ dollars, the demand is $ x = 7600 $ million bushels.

1. **Find the slope** $ m $ for the price-demand equation:
   \[
   m = \frac{p_2 - p_1}{x_2 - x_1} = \frac{2.36 - 2.25}{7600 - 7700} = \frac{0.11}{-100} = -0.0011
   \]

2. **Substitute one of the points to find $ b $:**
   Using the point $ (7700, 2.25) $:
   \[
   p = mx + b \implies 2.25 = -0.0011(7700) + b
   \]
   \[
   2.25 = -8.47 + b \implies b = 2.25 + 8.47 = 10.72
   \]

Thus, the price-demand equation is:
\[
p = -0.0011x + 10.72
\]

---

**(C) Equilibrium Point:**

The equilibrium occurs where the price-supply and price-demand equations are equal, so we set them equal to each other:
\[
0.000275x + 0.1875 = -0.0011x + 10.72
\]

1. Combine like terms:
   \[
   0.000275x + 0.0011x = 10.72 - 0.1875
   \]
   \[
   0.001375x = 10.5325
   \]

2. Solve for $ x $:
   \[
   x = \frac{10.5325}{0.001375} = 7662.18
   \]

3. Substitute $ x = 7662.18 $ into one of the original equations to find $ p $. Using the price-supply equation:
   \[
   p = 0.000275(7662.18) + 0.1875 = 2.2936 + 0.1875 = 2.4811
   \]

Thus, the equilibrium point is approximately:
\[
x = 7662.18 \text{ million bushels}, \quad p = 2.48 \text{ dollars}
\]

---

Would you like further details, or have any questions about these calculations? Here are 5 questions you can explore further:

1. How do we graph linear supply and demand functions on a coordinate plane?
2. What happens if the supply exceeds demand at a certain price?
3. How do changes in the slope affect the price-supply and price-demand relationships?
4. How can we interpret the equilibrium point in terms of market behavior?
5. What is the significance of the y-intercept in supply and demand equations?

**Tip**: The slope of a price-supply or price-demand equation indicates how sensitive the price is to changes in supply or demand.

At a price of $2.25 per bushel, the supply of a certain grain is 7500 million bushels, and the demand is 7700 million bushels. At a price of $2.36 per bushel, the supply is 7900 million bushels, and the demand is 7600 million bushels. (A) Find a price-supply equation of the form p = mx + b, where p is the price in dollars and x is the supply in millions of bushels. (B) Find a price-demand equation of the form p = mx + b, where p is the price in dollars and x is the demand in millions of bushels. (C) Find the equilibrium point.

Solve for the price-supply equation and the price-demand equation using the data points provided. Determine the equilibrium point where supply equals demand for a given grain at different price levels. Learn how to calculate slopes and solve linear equations.

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