At a price of 2.24 per bushel, the supply of a certain grain is 7100 million bushels and the demand is 7800 million bushels. At a price of 2.35 per bushel, the supply is 7500 million bushels and the demand is 7700 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 xis the demand in millions of bushels. (C) Find the equilibrium point. (D) Graph the price-supply equation, price-demand equation, and equilibrium point in the same coordinate system. (A) The price-supply equation is p=0.000275x+0.2875 (Type an exact answer.) (B) The price-demand equation is p=-0.0011x+10.808 (Type an exact answer.)

## Step 1: Find the price-supply equation<br />To find the price-supply equation, we need to determine the slope and y-intercept of the line that represents the relationship between price and supply.<br /><br />### Slope formula:<br />The slope of a line passing through two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:<br />\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]<br /><br />Using the given data points (7100, 2.24) and (7500, 2.35), we can calculate the slope:<br />\[ m = \frac{2.35 - 2.24}{7500 - 7100} = \frac{0.11}{400} = 0.000275 \]<br /><br />Now, we can use the point-slope form of a linear equation to find the equation of the line:<br />\[ p - p_1 = m(x - x_1) \]<br />Substituting the values, we get:<br />\[ p - 2.24 = 0.000275(x - 7100) \]<br />Simplifying, we have:<br />\[ p = 0.000275x + 0.2875 \]<br /><br />## Step 2: Find the price-demand equation<br />Similarly, we can find the price-demand equation by determining the slope and y-intercept of the line that represents the relationship between price and demand.<br /><br />Using the given data points (7800, 2.24) and (7700, 2.35), we can calculate the slope:<br />\[ m = \frac{2.35 - 2.24}{7700 - 7800} = \frac{0.11}{-100} = -0.0011 \]<br /><br />Using the point-slope form of a linear equation, we get:<br />\[ p - 2.24 = -0.0011(x - 7800) \]<br />Simplifying, we have:<br />\[ p = -0.0011x + 10.808 \]<br /><br />## Step 3: Find the equilibrium point<br />The equilibrium point is the point where the supply and demand curves intersect, meaning the price-supply equation is equal to the price-demand equation.<br /><br />Setting the two equations equal to each other, we have:<br />\[ 0.000275x + 0.2875 = -0.0011x + 10.808 \]<br />Solving for \(x\), we get:<br />\[ 0.001375x = 10.5215 \]<br />\[ x = 7640 \]<br /><br />Substituting this value back into either equation, we can find the equilibrium price:<br />\[ p = 0.000275(7640) + 0.2875 \]<br />\[ p = 2.11 \]<br /><br />So, the equilibrium point is (7640, 2.11).<br /><br />## Step 4: Graph the equations<br />To graph the price-supply equation, price-demand equation, and equilibrium point in the same coordinate system, we can plot the lines on a graph with the x-axis representing the supply/demand in millions of bushels and the y-axis representing the price in dollars.<br /><br />The price-supply equation will be a line with a positive slope passing through the points (7100, 2.24) and (7500, 2.35).<br />The price-demand equation will be a line with a negative slope passing through the points (7800, 2.24) and (7700, 2.35).<br />The equilibrium point will be a point on both lines, representing the intersection of the supply and demand curves.