EECM 3714
Lecture 2: Unit 2
Linear equations
Renshaw, Ch. 3
25 February 2022
,OUTLINE
• Linear equations and functions
• Simultaneous linear equations
• Two variables in two equations
• Three variables in three equations
• Economic applications
• Demand and supply analysis
• National Income
• IS-LM analysis
, EQUATIONS AND IDENTITIES
Manipulating equations
• Equation manipulation is needed in order to extract useful information.
• When manipulating equations, any operation must be performed on both sides of the
equation.
• In most equation manipulation our objective is to isolate the unknown, x, on one side of
the equation.
If any operation performed on the whole of both sides of an equation, the equation remains a
true statement. Examples:
If x = 5, then x + 100 = 105 ; if a = b, then 2a = 2b
By "operation" we mean add, subtract, multiply, divide or raise to any power (but not power
zero).
Variables and parameters
An equation may contain one or more unknown values, called variables (x, y, z) and one or
more known values, called parameters (coefficients or constants, a, b, c)
, GENERAL LINEAR EQUATION
• Equation is linear if none of the variables that appear in it is raised to any pow
except power 1. (x = x1)
• Consider 𝑎𝑥 + 𝑏 = 𝑐, in which 𝑥 is the only variable, while 𝑎; 𝑏 and 𝑐 are
constants and/or parameters.
𝑐−𝑏
• The solution to this equation is 𝑥 =
𝑎
• Check by replacing x with its solution:
( )
a c a−b + b = c c − b + b = c (an identity)
• Note that any equation becomes an identity when we replace the unknown w
its solution.
• Linear function → Constant relationship between 𝑥 and 𝑦
Lecture 2: Unit 2
Linear equations
Renshaw, Ch. 3
25 February 2022
,OUTLINE
• Linear equations and functions
• Simultaneous linear equations
• Two variables in two equations
• Three variables in three equations
• Economic applications
• Demand and supply analysis
• National Income
• IS-LM analysis
, EQUATIONS AND IDENTITIES
Manipulating equations
• Equation manipulation is needed in order to extract useful information.
• When manipulating equations, any operation must be performed on both sides of the
equation.
• In most equation manipulation our objective is to isolate the unknown, x, on one side of
the equation.
If any operation performed on the whole of both sides of an equation, the equation remains a
true statement. Examples:
If x = 5, then x + 100 = 105 ; if a = b, then 2a = 2b
By "operation" we mean add, subtract, multiply, divide or raise to any power (but not power
zero).
Variables and parameters
An equation may contain one or more unknown values, called variables (x, y, z) and one or
more known values, called parameters (coefficients or constants, a, b, c)
, GENERAL LINEAR EQUATION
• Equation is linear if none of the variables that appear in it is raised to any pow
except power 1. (x = x1)
• Consider 𝑎𝑥 + 𝑏 = 𝑐, in which 𝑥 is the only variable, while 𝑎; 𝑏 and 𝑐 are
constants and/or parameters.
𝑐−𝑏
• The solution to this equation is 𝑥 =
𝑎
• Check by replacing x with its solution:
( )
a c a−b + b = c c − b + b = c (an identity)
• Note that any equation becomes an identity when we replace the unknown w
its solution.
• Linear function → Constant relationship between 𝑥 and 𝑦
