## algebraic problem: logic behind a graphing dilemma?

Quadratic equations and inequalities, variation equations, function notation, systems of equations, etc.
jon80
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Joined: Fri Sep 14, 2012 7:01 am
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### algebraic problem: logic behind a graphing dilemma?

I am trying to find the point of intersection of the following graphs, and, wondering whether it may help me with the logic behind a graphing dilemma that I have with an economic problem, because somehow I am thinking that the point of intersection for the production possibility curve represents something, but I am not sure what. My tutor is not very helpful via email.

Problem 1 What is the point of intersection of the following functions?
f(x) = 2x + 3
g(x) = 0.5x + 7

When working it out manually the point of intersection is where f(x) = g(x), and, so far I have figured out that this happens where x = 6 although I don't know how to work out the value of y. Initially I thought I would solve 2x + 3 = y, but, this is incorrect according to the graphing software I use to check my answer, since the point of intersection is expected to be where x = 1.57 and y = 6.13, so how do I work this out using algebra?

Problem 2.
Two economies are identical: In each, a unit of labour can make either 3 units of y or 1 unit of x (or
any linear combination of them). A unit of capital can make either 3 units of y or 3 units of x (or
any linear combination of them). In each economy there are 100 units of labour, and 50 units of
capital. Both labour and capital are needed in theproduction process. If economy A chooses to
consume efficiently at a point where y=65 while economy B chooses to consume efficiently at a
point where x=65, the economies will not trade. True or false? Explain.
LSE May 2012 Zone B Q.1a

anonmeans
Posts: 84
Joined: Sat Jan 24, 2009 7:18 pm
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### Re: algebraic problem: logic behind a graphing dilemma?

I am thinking that the point of intersection for the production possibility curve represents something, but I am not sure what.
Intersections are where things are equal. So maybe if the equations are for costs per unit the intersection stands for where the unit costs are the same.
Problem 1 What is the point of intersection of the following functions?
f(x) = 2x + 3
g(x) = 0.5x + 7

When working it out manually the point of intersection is where f(x) = g(x), and, so far I have figured out that this happens where x = 6 although I don't know how to work out the value of y.
You get y by plugging in x. Since the intersection is where they're equal you can plug x into either one of the equations. How did you get x=6? How did you get x=1.57, y=6.13? Check it:

2(1.57)+3=6.14
0.5(1.57)+7=7.785

They're not even close.
Problem 2. Two economies are identical: In each, a unit of labour can make either 3 units of y or 1 unit of x (or any linear combination of them). A unit of capital can make either 3 units of y or 3 units of x (or any linear combination of them). In each economy there are 100 units of labour, and 50 units of capital. Both labour and capital are needed in the production process. If economy A chooses to consume efficiently at a point where y=65 while economy B chooses to consume efficiently at a point where x=65, the economies will not trade. True or false? Explain.
I don't know the economics stuff. What is the math meaning of "consuming efficiently"? What are the rules for whether or not they'll trade?