In supply and demand problems, y is the number of items the supplier will produce
or the consumers will buy if the price of the item is x dollars. Where our supply equation
is y equals six x plus five hundred thirty and our demand equation is y equals negative eight x
plus seven hundred forty. We first want to find the intersection point of the two lines, then
determine what the selling price is when the supply and demand are in equilibrium, and then
finally we want to find the amount of items in the market when supply and demand are in equilibrium.
Let's begin by looking at the graph of these two linear equations on the coordinate plane.
Notice the supply equation is graphed here in red and notice how because the
slope is positive we have an increasing linear function which means as the price increases,
y the number of items supplied to the market also increases and then the demand equation is graphed
here in blue because it has a negative slope we have a decreasing linear function which means
as the price increases or as x increases the number of items consumers will buy will decrease.
The equilibrium point is when the supply and demand equations are equal,
which would be at this point of intersection and our first question was to find this point
of intersection so we'll do this algebraically by solving this as a system of equations.
So let's begin by doing that. If we want to solve this as a system,
the preferred method would probably be to use substitution where if y equals six x plus
five hundred thirty we can substitute six x plus five hundred thirty for y in the second equation
by doing this we'll have one equation with one variable and then we can solve for x.
So if we make this substitution, we would have six x plus five hundred thirty equals
negative eight x plus seven hundred forty. So for the first step, we want the variable
terms on one side of the equation. Let's go ahead and add eight x to both sides
so this would be zero so now we have fourteen x plus five hundred thirty equals seven hundred
forty. And now we'll isolate the variable term by subtracting five hundred thirty on both sides.
So we have fourteen x equals
this would be two hundred ten and now we divide both sides by fourteen.
So we have x equals fifteen
which means for the point of intersection we now know the x coordinate is fifteen. So to
find the y coordinate we'll substitute fifteen for x in either of the original equations.
So if we use the first equation again we now know that x equals fifteen
which means y must equal six times fifteen plus five hundred thirty.
Well six times fifteen would be ninety so we have y equals ninety plus five hundred thirty or y
equals six hundred twenty. Now if we use the second equation we would still get y equals
six hundred twenty and again this would be the y coordinate of the point of intersection.
Let's go back to the graph to verify this. Notice how the x coordinate of the point of
intersection is fifteen and the y coordinate of the point of intersection is six hundred twenty.
So going back to our three questions we just found the point of intersection to be fifteen comma
six hundred twenty. Next what is the selling price when the supply and demand are in equilibrium?
That would be the x coordinate of the point of intersection so the selling price is fifteen when
the supply and demand are in equilibrium and then for the last question what is the amount of items
in the market when the supply and demand are in equilibrium? That would be the y coordinate of the
point of intersection which is six hundred twenty. So the number of items is six hundred twenty again
when the supply and demand are in equilibrium. I hope you found this helpful.