Linking graphs and systems of equations Graph of a linear equation Graphical solutions to systems

Linking graphs and systems of equations Today, we introduce the concept of a graph to find a solution to a system of equations As for last week we will examine systems of 2 equations/unknowns This gives us another way of solving systems of equations Often useful in economics (and for checking results to your calculations) However, it also allows us to understand better identification problems

Linking graphs and systems of equations Graph of a linear equation Solving a system of 2 equations graphically Graphs, systems and identification

Graph of a linear equation The general notation for a linear equation is given by: Where (x,y) are unknowns and (a,b) are parameters. Lets imagine that a = 0.5 and b = 5 What is the graph of the function

Graph of a linear equation Going back to week 4, we first need some axes Vertical axis ‘y’ axis Horizontal axis ‘x’ axis

Graph of a linear equation This allows us to graph the function x → y=0.5x + 5 x y

Graph of a linear equation In order to do so, we need to know the values that the function takes for all x’s. This is done with the ‘variation table’ For a linear function you only need 2 points The x=0 point often provides an easy start For non-linear functions, this is not the case! x=0x=1x=2x=3x=4x=5 y=0.5x

Plotting the data from the variation table allows you to obtain the graph of the function: Graph of a linear equation x y

Linking graphs and systems of equations Graph of a linear equation Solving a system of 2 equations graphically Graphs, systems and identification

Solving a system of equations graphically We will base this analysis on the supply and demand example we saw last week: We already have the solution for P and Q, which we worked out analytically. The system solves for P = 400 and Q = 800. We will now solve the system graphically to show that the solution point is the same.

Solving a system of equations graphically Step 1: modify the system to express the equations as functions in your graphical space In economics, price is on the vertical axis, quantity on the horizontal one Q P

Solving a system of equations graphically Step 1 (cont’d) : The system becomes Step 2 : Draw each function in the available space

Step 3 : The solution is given by the coordinates of the intersection of the 2 functions P = 400 and Q = 800 ! Solving a system of equations graphically Q P

Linking graphs and systems of equations Graph of a linear equation Solving a system of 2 equations graphically Graphs, systems and identification

The graph of a system of equations allows us to find the solutions to a system of 2 equations and 2 unknowns, but it also allows us to understand why certain systems don’t have solutions Example : one of the systems you had as an exercise… Why can’t it be solved ?

Graphs, systems and identification Rearranging the system expressing y as a function of x: Let’s see what the graph of this system looks like…

Graphs, systems and identification The functions are parallel, no intersection exists They are said to be “co-linear” x y

Graphs, systems and identification This is similar to having twice the same equation ! This means that for a system to have a solution, you need 2 independent (different) equations If they are co-linear, no solution exists because you have twice the same information… What about this second case :

Graphs, systems and identification This time there are too many solutions !! Again, there is no defined “single” solution… x y

Graphs, systems and identification The graphical approach clarifies why you need exactly N equation for N unknowns: The system is “under-indentified”: If the number of equations is smaller than the number of variables Or if some equations are co-linear The system is “over-indentified”: If there are more equations than unknowns The system is “just-indentified”: If the number of (independent) equations is equal to the number of unknowns