To determine the equilibrium point where supply and demand are equal, we need to solve the system of equations given by the supply and demand equations. Specifically, we need to find the value of [tex]\( x \)[/tex] (price) where the number of items supplied equals the number of items demanded ([tex]\( N \)[/tex]).
The equations provided are:
1. Supply equation: [tex]\( N = 3x + 511 \)[/tex]
2. Demand equation: [tex]\( N = -2x + 676 \)[/tex]
At the equilibrium point, the supply equals the demand. Therefore:
[tex]\[ 3x + 511 = -2x + 676 \][/tex]
We'll solve for [tex]\( x \)[/tex] step by step:
1. Combine the terms involving [tex]\( x \)[/tex] and the constant terms:
[tex]\[ 3x + 2x = 676 - 511 \][/tex]
2. Simplify the equation:
[tex]\[ 5x = 165 \][/tex]
3. Solve for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{165}{5} \][/tex]
[tex]\[ x = 33 \][/tex]
Now that we have the equilibrium price [tex]\( x \)[/tex], we substitute this value back into either the supply or demand equation to find the equilibrium quantity [tex]\( N \)[/tex].
Using the supply equation [tex]\( N = 3x + 511 \)[/tex]:
[tex]\[ N = 3(33) + 511 \][/tex]
[tex]\[ N = 99 + 511 \][/tex]
[tex]\[ N = 610 \][/tex]
So, the equilibrium price (selling price) when supply and demand are in equilibrium is [tex]\( x = 33 \)[/tex] dollars, and the number of items in the market is [tex]\( N = 610 \)[/tex].
To summarize:
1. The two lines intersect at the point: [tex]\((33, 610)\)[/tex].
2. The selling price per item when supply and demand are in equilibrium is [tex]\(\$ 33\)[/tex].
3. There are 610 items in the market when supply and demand are in equilibrium.
