supply and demand: for a certain commodity the supply equation is given by S=2p+5

at a price o $1, there is a demand for 19 units of the commodity. If the demand equation is linear and the market price is $3, find the demand equation

To find the demand equation, we need to use the given information about the price and demand for the commodity.

We know that at a price of $1, the demand for the commodity is 19 units. Let's denote the demand as D and the price as p.

So, according to the given information, when p = $1, D = 19.

Now, we also know that the demand equation is linear. In linear equations, the general form is y = mx + b, where y is the dependent variable (in this case, the demand), x is the independent variable (in this case, the price), m is the slope of the line, and b is the y-intercept.

For our case, we need to determine the values of m and b to derive the demand equation.

We are given one point on the line, (p = 1, D = 19). So, we can substitute these values into the linear equation:

19 = m(1) + b

Simplifying this equation, we get:

19 = m + b ---- (1)

To find the value of m (the slope), we need another point. We are given that at a price of $3, but we do not know the corresponding demand. Let's denote this demand as D2.

Using the supply equation S = 2p + 5, we can find the supply at p = $3:

S = 2(3) + 5
S = 6 + 5
S = 11

Since supply equals demand in equilibrium, we can equate this to D2:

D2 = 11

Now, we can substitute this point (p = 3, D = 11) into the linear equation:

11 = m(3) + b

Simplifying this equation, we get:

11 = 3m + b ---- (2)

Now, we have a system of two equations with two variables:

Equation (1): 19 = m + b
Equation (2): 11 = 3m + b

To find the values of m and b, we can solve this system of equations using any available method, such as substitution or elimination.

Using the elimination method, we can subtract Equation (1) from Equation (2) to eliminate b:

(11 - 19) = (3m + b) - (m + b)
-8 = 3m - m
-8 = 2m

Dividing both sides by 2, we get:

-4 = m

Now, substitute this value of m back into Equation (1) to find b:

19 = (-4) + b
19 + 4 = b
23 = b

Therefore, we have found the values of m and b. The demand equation (linear) is given by:

D = -4p + 23

So, the demand equation for the commodity is D = -4p + 23.