a. Suppose a market research company finds that at a price of p = $30, they would sell x = 58 tiles each month. If they lower the price to p = $18, then more people would purchase the tile, and they can expect to sell x = 70 tiles in a month’s time. Find the equation of the line for the demand equation.

(58,30), (70,18),

m = (18-30) / (70-58) = -12 / 12 = -1.

Y = mx + b = 30,
-1*58 + b = 30,
b = 88.

Eq: P = -X + 88

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To find the equation of the line for the demand equation, we can use the given information about the price and quantity demanded.

Let's first assign variables to the price (p) and the quantity demanded (x). We'll use p as the independent variable and x as the dependent variable.

From the information provided, we have two data points: when the price is $30, the quantity demanded is 58, and when the price is $18, the quantity demanded is 70.

We can use these two points to find the equation of the line using the point-slope form of a linear equation.

The point-slope form of a linear equation is given by:

y - y₁ = m(x - x₁),

where (x₁, y₁) is a point on the line and m is the slope of the line.

Let's use the point (30, 58) as our first point.

So, x₁ = 30 and y₁ = 58.

Now let's find the slope, m, using the second point (18, 70).

The slope, m, can be calculated as:

m = (y₂ - y₁) / (x₂ - x₁).

Plugging in the values for the second point, we have:

m = (70 - 58) / (18 - 30) = 12 / (-12) = -1.

Now we have the slope, m = -1, and a point on the line, (30, 58).

Using the point-slope form, we can write the equation as:

x - x₁ = m(p - p₁).

Plugging in the values:

x - 58 = -1(p - 30).

Simplifying further, we get:

x - 58 = -p + 30.

Re-arranging the equation gives us the final equation of the demand line:

x = -p + 88.

So, the equation of the line for the demand equation is x = -p + 88.