# Homework Help: Linear Equation (supply and demand line)

1. Sep 24, 2010

### nukeman

1. The problem statement, all variables and given/known data

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 ?

2. Relevant equations

3. 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

2. Sep 24, 2010

### Mentallic

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...

3. Sep 24, 2010

### nukeman

4. Sep 24, 2010

### Mentallic

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...

5. Sep 24, 2010

### HallsofIvy

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 $$a= \frac{y_1- y_0}{x_1- x_0}$$.

6. Sep 24, 2010

### Mentallic

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

7. Sep 24, 2010

### nukeman

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

8. Sep 24, 2010

### Mentallic

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

9. Sep 24, 2010

### nukeman

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 :)

10. Sep 24, 2010

### Mentallic

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...

11. Sep 24, 2010

### nukeman

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!

12. Sep 24, 2010

### Mentallic

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

13. Sep 24, 2010

### nukeman

HOw? -4p + 3

14. Sep 24, 2010

### Mentallic

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

15. Sep 24, 2010

### nukeman

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

16. Sep 24, 2010

### Mentallic

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

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

17. Sep 24, 2010

### nukeman

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

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

18. Sep 24, 2010

### Mentallic

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.