# Linear Equation (supply and demand line)

• nukeman
But I think you should stick around and ask some questions in the algebra section, you'll learn a lot that way :smile:In summary, the conversation discusses a problem where the supply equation is given by S=2p+5 and the demand equation is linear. It is given that at a price of $1, there is a demand for 19 units of the commodity. To find the demand equation, the market price of$3 is substituted into the supply equation to find the corresponding supply value of 11. This point, along with the point (1,19) given in the problem, is used to find the demand equation D=ap+b, where a and b are the unknown coefficients. The solution provided in the conversation
nukeman

## Homework Statement

Question:

For a certain commodity the supply equation is given by:

S = 2p + 5

At a price of $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 ?

## The Attempt at a Solution

I am not sure the steps I have to take to solve this. Can anyone help?

I know first I would do this:

At p = 3, S = 2(3) + 5 = 11

My economics is quite rusty - actually, inexistent would be a more accurate term - but this is maths so I reckon I can smudge the answer together

So we have that $$S=2p+5$$ and we are trying to find the linear equation of the demand curve, given by $$D=ap+b$$. We are searching for the values of a and b to complete the answer.

It is given that (1,19) satisfies the demand curve, thus, $$19=a+b$$ (1)
You accidentally subbed these values into your supply curve.

When the market price is $3, I think this means that the supply and demand curves are equal? Correct me if I'm wrong here. If this is true, then the curves $$S=D$$, thus $$2p+5=ap+b$$ and we're given that at this value, the market price is$3, thus...

That's another way of doing the problem. My approach was to let the demand line be an equation D=ap+b and then I would find a and b, the solution in that link is to find two points that lie on the demand line and then find the equation of the line between those two points.

What exactly do you need clarification on because it gave you the entire solution...

nukeman said:

(its number 2)

http://web.viu.ca/pughg/Fall2009/math111F09N01/math111F09N01q2Sol.pdf
Essentially they are just using a formula for a straight line: The equation of a line through $(x_0, y_0)$ and $(x_1, y_1)$ is given by
$$y= \frac{y_1- y_0}{x_1- x_0}(x- x_0)+ y_0$$

You can derive that formula by using the method Mentallic used: any (non-vertical) line can be written as y= ax+ b. If the line goes through $(x_0, y_0)$ then we must have $y_0= ax_0+ b$. If the line also goes through $(x_1, y_1)$ then we must also have $y_1= ax_1+ b$. Subtracting the first equation from the second eliminates b: $y_1- y_0= ax_1- ax_0= a(x_1- x_0)$. Now divide both sides by $x_1 - x_0$ to get $$a= \frac{y_1- y_0}{x_1- x_0}$$.

Yeah... I think that's stepping a little out of the range of nukeman's understanding here halls.

Mentallic said:
That's another way of doing the problem. My approach was to let the demand line be an equation D=ap+b and then I would find a and b, the solution in that link is to find two points that lie on the demand line and then find the equation of the line between those two points.

What exactly do you need clarification on because it gave you the entire solution...

Im just having a hard time understanding all the steps in that solution.

nukeman said:
Im just having a hard time understanding all the steps in that solution.

Ok then, do you understand the steps in my solution?

Mentallic said:
Ok then, do you understand the steps in my solution?

Yes I do, thanks!

Is there anyway you can explain the steps given in the solution I gave you?

But thanks for your solution, seems easier :)

Ok that's good to hear

Sure thing, it's really the same idea, just once you have the necessary data together they took a different approach to finding the answer.

S=2p+5

and we are told that $3 is the market price which means the supply and demand curve intersect at this point. So S=2(3)+5=11 therefore we have a point (3,11) which lies on the demand line (and on the supply line). This is our first point, now we need a second and then we just 'connect the dots' to make a line between them to give us the demand curve. The info also tells us that at the price of$1, the demand is 19 units, so (1,19) also lies on the demand curve.

These are the two points we needed, now we just formulae and such to find the equation of the curve.

All I really did was reword their solution... lol...

Right...Think I am getting it now (thank you!)

Can you show me how they get this answer ( I know, just want to see the steps to make sure I got it)

answer is: D - 11 = -4 (p-3)

P.S - Mentallic, REALLY appreciate the help!

Well, can you expand -4(p-3)?

Mentallic said:
Well, can you expand -4(p-3)?

HOw? -4p + 3

The rule is a(b+c)=ab+ac

So how would you apply that to: -4(p-3) ?

a=-4, b=p, c=-3

Are you studying economics or mathematics? Because this is elementary maths that is definitely a prerequisite for economics.

Mentallic said:
a=-4, b=p, c=-3

Are you studying economics or mathematics? Because this is elementary maths that is definitely a prerequisite for economics.

Been out of school for a while, just starting Math again.

I knwo I know, I am not good at math right now :(

Ahh ok so you're studying maths. Well since you don't remember how to expand, you should definitely start learning basic algebra before you get into coordinate geometry. You can rush through it of course, it will save you time and heartache in the end.

## 1. What is a linear equation in terms of supply and demand?

A linear equation in terms of supply and demand is a mathematical representation of the relationship between the quantity of a good or service that is supplied and the quantity of that good or service that is demanded. It is expressed as Qs = mP + b, where Qs is the quantity supplied, m is the slope of the line, P is the price of the good or service, and b is the y-intercept.

## 2. How do you graph a linear equation for supply and demand?

To graph a linear equation for supply and demand, plot the y-intercept on the vertical axis and use the slope to determine the rest of the points on the line. The slope represents the change in quantity over the change in price. Once you have two or more points, connect them with a straight line to create the supply or demand curve.

## 3. What does the slope of the supply and demand line represent?

The slope of the supply and demand line represents the change in quantity over the change in price. It is also known as the marginal cost or marginal benefit. A steeper slope indicates a greater change in quantity for a given change in price, while a flatter slope represents a smaller change in quantity for the same change in price.

## 4. How does a change in price affect the supply and demand line?

A change in price will cause a movement along the supply or demand line. If the price increases, the quantity supplied also increases along the supply line. On the other hand, the quantity demanded decreases along the demand line. If the price decreases, the opposite occurs. However, a change in price does not shift the supply and demand line itself.

## 5. What factors can cause a shift in the supply and demand line?

A shift in the supply and demand line can be caused by factors such as changes in consumer preferences, changes in technology, changes in the cost of inputs, changes in the number of producers, and changes in government policies. These shifts can result in a new equilibrium point where the quantity supplied and demanded are equal at a different price and quantity than before.

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