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


This gives us another way of solving
systems of equations


As for last week we will examine systems of 2
equations/unknowns
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:
y  ax  b


Where (x,y) are unknowns and (a,b) are
parameters.
Lets imagine that a = 0.5 and b = 5
y  0.5 x  5

What is the graph of the function y  f  x 
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
y
y  0.5 x  5
x
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’
x=0 x=1 x=2 x=3 x=4 x=5
y=0.5x+5

5.5
6
6.5
7
7.5
For a linear function you only need 2 points


5
The x=0 point often provides an easy start
For non-linear functions, this is not the case!
Graph of a linear equation

Plotting the data from the variation table allows
you to obtain the graph of the function:
y
y  0.5 x  5
x
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.
Q s  400  P
 d
Q  1600  2 P


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
P
Q
Solving a system of equations graphically

Step 1 (cont’d) : The system becomes
Q  400  P
 d
Q  1600  2 P
s

 P  400  Q s

d
 P  800  0.5Q
Step 2 : Draw each function in the available
space
Solving a system of equations graphically
P
 P  400  Q s

d
P

800

0.5
Q

Supply
1200
P  400  Q s
1000
800
600
400
Demand
200
P  800  0.5Q d
Q
200 400 600 800 1000

1600
Step 3 : The solution is given by the coordinates
of the intersection of the 2 functions

P = 400 and Q = 800 !
Linking graphs and systems of equations
Graph of a linear equation
Solving a system of 2 equations
graphically
Graphs, systems and identification
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…
x  5y  8

2 x  10 y  12

Why can’t it be solved ?
Graphs, systems and identification

Rearranging the system expressing y as a
function of x:
x  5y  8

2 x  10 y  12

 y  1.6  0.2 x

 y  1.2  0.2 x
Let’s see what the graph of this system looks
like…
Graphs, systems and identification
y
2.4
2.2
 y  1.6  0.2 x

 y  1.2  0.2 x
2
1.8
1.6
1.4
1.2
x
1

2
3
4
5
The functions are parallel, no intersection exists

They are said to be “co-linear”
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 :
 y  12  2 x

 y  5  0.5 x
 y  10  x

Graphs, systems and identification
y
 y  12  2 x

 y  5  0.5 x
 y  10  x

12
10
8
6
4
2
x
2

4
6
8
10 12 14
16 18
This time there are too many solutions !!

Again, there is no defined “single” solution…
Graphs, systems and identification


The graphical approach clarifies why you need
exactly N equation for N unknowns:
The system is “under-indentified”:



The system is “over-indentified”:


If the number of equations is smaller than the
number of variables
Or if some equations are co-linear
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