At a price of $110,
there is a demand for 1,770 items
and a supply of 990 items.
At a price of $210 there is a demand for 1,170 items
and a supply of 1,890 items.
Assuming supply and demand are linear,
find the equilibrium price and quantity.
Let's first record the given information.
We know at a price of $110,
the demand is 1,770 items, and the supply is 990 items.
At a price of $210,
we know the demand is 1,170 items,
and the supply is 1,890 items.
Because we know the demand and supply are linear,
we will now list two points on the demand function,
two points on the supply function,
and then use those points to determine the linear equation
for both the demand and the supply,
and we'll find the points
in the form of quantity comma price.
Beginning with two points on the demand function,
we know the demand is 1,770 when the price is $110.
This gives us the ordered pair 1,770 comma 110.
And we also know the demand is 1,170 when the price is $210.
This gives us the second point of 1,170 comma 210.
And now, let's find two points on the supply function,
again, in the form of quantity comma price.
The supply is 990 items when the price is $110,
which gives us the ordered pair 990 comma 110.
We also know the supply quantity is 1,890
when the price is $210.
This gives us a second ordered pair of 1,890 comma 210.
And now, let's use these points to find in the equation
for the demand function, as well as the supply function,
because we know the functions are linear.
So, using the two points on the demand function,
we will now find the slope, where the slope
is normally equal to the change of y
divided by the change of x.
In our case, the slope is equal to the change in price
divided by the change in the quantity.
The change in price is equal to 210 minus 110.
The change in the quantity is 1,170 minus 1,770.
Simplifying, we have the slope m is equal to 100
divided by negative 600, which simplifies to negative 1/6.
From here, we'll use the point-slope form of a line
where m is the slope and x sub one, y sub one
is one point in the line
in order to find our equation for the demand.
Of course, instead of x and y we'll be using q and p.
So, our slope is negative 1/6,
and let's use the first point.
This gives us the equation p minus 110
is equal to negative 1/6
times the quantity q minus 1,770.
And now, we solve for p.
Distributing the negative 1/6, we have negative 1/6 q.
And then, negative 1/6 times negative 1,770
is equal to positive 295, giving us plus 295.
And now, to solve for p, we add 110 to both sides.
Simplifying, we have the price
is equal to negative 1/6 times the quantity,
and then plus 405.
And this is our demand function.
And now, we do the same for the supply function,
beginning with the slope.
The slope m, again, is equal to the change in price
divided by the change in the quantity,
which is equal to 210 minus 110
divided by 1,890 minus 990.
Simplifying, the slope is equal to 100 divided by 900,
which simplifies to 1/9.
And now, let's find the equation again
in point-slope form, and then solve for p.
And again, let's use the first point.
We have p minus 110 equals m, which is 1/9,
times the quantity q minus 990.
Simplifying, we distribute 1/9, which gives us 1/9 q.
And then, 1/9 times negative 990
is equal to negative 110, giving us minus 110.
Solve for p.
Again, we add 110 to both sides.
Simplifying, we have p equals,
on the right we just have 1/9 q.
Notice negative 110 plus 110 is equal to zero.
And now, we have the supply function.
In order to find the equilibrium price and quantity,
we now solve this as a system of equations,
and let's do this on the next slide.
Let's use the method of substitution.
If p is equal to 1/9 q,
we can substitute p in the demand function with 1/9 q.
This gives us the equation 1/9 q
equals negative 1/6 q plus 405.
Let's get the q terms on the same side
by adding 1/6 q to both sides, which gives us 1/9 q
plus 1/6 q equals 405.
We now need to add the fractions on the left.
The least common denominator is going to be 18.
We multiply the numerator and denominator of 1/9 by two,
and multiply the numerator and denominator of 1/6 by three.
And then, we have a common denominator of 18.
We have 2/18 plus 3/18, which is equal to 5/18.
This gives us 5/18 q is equal to 405.
To solve for q, we multiply both sides
by the reciprocal of 5/18, which is 18/5.
Simplifying, on the left, we just have one times q, or q.
On the right, five and 405 share a common factor of five.
There is one five in five, and 81 fives in 405.
The quantity is equal to 81 times 18,
which is equal to 1,458.
And now, we need to find the price.
The price is equal to 1/9 times q,
which we now know is 1,458,
which equals 162.
Now, we know the equilibrium price is $162,
and we know the equilibrium quantity is 1,458 items.
And let's go ahead and verify this graphically.
The demand function has graphed in blue,
the supply function is graphed in red.
And notice how the point of intersection
is 1,458 comma 162,
which does verify the equilibrium quantity
and the equilibrium price.
I hope you found this helpful.