Assignment 5: Solving Systems of Equations



Mathematical Economics: Practice Questions on Solving Systems of Equations

Two Equation Systems

Basic

1. Solve for x and y in the following equations

[pic]

Solve first equation for y

[pic]

Then substitute for y in second equation and find x

x = 7

Find y

[pic]

2. Solve for x and y in the following equations

[pic]

Solve first equation for y

[pic]

Substitute for y in second equation and find x

[pic]

Find y

[pic]

Market Equilibrium

Solve for the equilibrium price and quantity in the following equations:

1. Demand: [pic]

Supply: [pic]

Set supply = demand and solve for p

[pic]

Substitute p into either supply or demand and solve for q. I will substitute into supply.

[pic]

2. Demand: [pic]

Supply: [pic]

Set supply = demand and solve for p

[pic]

Substitute p into either supply or demand and solve for q. I will substitute into supply.

[pic]

Three Equation Systems

Basic

Solve for x, y, and z from the following equations:

1. [pic]

The logic to solve a three equation system is to eliminate one variable and one equation. This leaves one with a two equation system. Then eliminate and other variable and equation, which leaves one with a one equation/one variable system. Then solve for that variable, and work backward to recover the solution to the other variables. Because it does not matter which equation or variable you start with, there are many different ways to apply this method of elimination of variables. The way I did was to solve the second equation for z and substitute into the third equation. This gives me two equations (1st and 3rd) and two variables (x, y). I then solved the first equation for y and substituted into the third. Then I solved the third for x. After finding x I worked backward to recover y and z. The final answer is as follows:

[pic]

2. [pic]

I solved this by solving for z from the 3rd equation and substituting into the 2nd equation. I then solved the first equation for x and substituted into the 2nd equation. I then solved for y, and worked backward to get final solutions for x and z. The final answer is as follows:

[pic]

NOTE: Because one can simplify equations in different ways your solutions may look different from the above.

National Income

Solve for the equilibrium national income in the following equations:

1. [pic]

While we did not do these in lecture, they are simple and provide some practice in applying solutions to systems to a different problem. Though this is three equation system with three unknown variables (Y, C, I) it is easy to solve by substituting for C and I in equation 1. You then solve for Y and can easily recover C and I if you want. NOTE: G is called an exogenous variable representing government policy. It is left in the same form as the variables (that is, capitalized) because it is important to other applications of this model.

[pic]

2. [pic]

As above this is easy to solve. We have added another equation and variable. G is now an “endogenous” variable which we can solve for. It is given by the simple equation T = G, where T represents taxes. Here substitute for C, I, and G in equation 1. We then have

[pic]

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