Lecture 2: Unit 2, Linear Equations
5/2/2016
Outline
1
Linear equations and functions
2
Simultaneous linear equations
Two variables in two equations
Three variables in three equations
3
Economic applications
Demand and supply analysis
National Income
ISLM analysis
Objectives
•
Manipulate and solve linear equations
•
Sketch the graph of a linear function
•
Solve systems of simultaneous linear equations
Literature
Renshaw, ch. 3
General form of linear equation
Consider
ax
+
b
=
c
, in which x is the only variable, while
a
,
b
and
c
are constants and/or parameters The solution to this equation is
x
=
c

b
a
•
Linear function: Constant relationship between
x
and
y
The function
y
=
ax
+
b
Consider the linear function
y
=
ax
+
b
In this function:
•
y
is the dependent variable
•
x
is the independent variable
•
a
is the slope
•
Slope: Shows the change in
y
for oneunit change in
x
, i.e.
a
=
Δ
y
Δ
x
•
y
intercept: Shows the value of
y
if
x
is 0
•
x
intercept: The
x
intercept shows the value of
x
if
y
is 0, i.e.
x
=

b
/
a
Inverse functions and Implicit vs. Explicit functions
•
Inverse function: if
y
=
ax
+
b
then
x
=
y

b
a
is the inverse
function
•
Note that for the inverse function, the slope is 1
/
a
while the
x
intercept is

b
/
a
•
Explicit function:
y
=
ax
+
b
. It’s clear which variable is the
dependent, and which is the independent
•
Implicit function:
ax
+
by

c
= 0. It’s not clear which
variable is the dependent, and which one the independent
Simultaneous equations: 2 by 2
The solution to systems of linear equations involves finding values
for all of the variables that satisfy all of the equations in the
system simultaneously.
Consider the following example: Solve for
x
and
y
in the following
system of equations
8
x
+ 4
y
= 12

2
x
+
y
= 9
There are two ways to solve this problem:
•
Substitution
•
Elimination
Solution via substitution
•
Take the second equation, and make
y
the subject:
y
= 2
x
+9
•
Now, plug this expression for
y
into the first equation:
8
x
+ 4(2
x
+ 9) = 12
•
Next, simplify: 8
x
+ 8
x
+ 36 = 12
•
∴
16
x
=

24
•
∴
x
=

24
16
=

3
2
=

1
.
5
•
Plug this solution into the expression for
y
:
y
= 2(

1
.
5) + 9 = 6
Solution via elimination
•
Again, solve for
x
and
y
given
•
8
x
+ 4
y
= 12
•

2
x
+
y
= 9
•
First, decide which variable you want to get rid of (let’s
choose
y
)
•
Now, multiply first equation by 1; multiply the second
equation by 4
•
The system then becomes:
•
8
x
+ 4
y
= 12
•

8
x
+ 4
y
= 36
•
Note that both equations contain the term 4
y
Solution via elimination 2
•
Now subtract the second equation from first equation :
•
8
x
+ 4
y

(

8
x
+ 4
y
) = 12

36
•
This yields 16
x
=

24
∴
x
=

1
.
5
•
Now plug
x
=

1
.
5 into any of the equations above (let’s
choose the first equation here):
•

2(1
.
5) +
y
= 9
∴
y
= 6
•
Note that these answers are the same (as they should be!)
Simultaneous equations: 3 variables in 3 equations
•
We can again solve for these using either elimination or
substitution
•
There are just a few more steps to follow
•
Consider the following system of equations:
•
x
+
y

z
=

4
•
2
x

y

z
= 0
•
3
x

4
y

z
= 0
Substitution
•
Rewrite the first equation, making
z
the subject:
x
+
y
+ 4 =
z
.
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 Summer '20
 Linear Equations, Supply And Demand, Elementary algebra