MATH 170B
NATHANIEL HALL
,UNIT ONE
, Mathematical Structure II Spring 2025
Section 1.1: Solutions and Elementary Operations
Lecturer: Mr. Hall Math 170B
Motivation
Find all solutions of the (linear) equation in one variable:
ax = b
Solution
• If a →= 0, there is a unique solution x = b/a.
• Else if a = 0 and
1) b →= 0, there is no solution. >
- contradiction
2) b = 0, there are infinitely many solutions, in fact any x ↑ R is a solution. stautology
This is a complete description of all possible solutions of ax = b.
Can we do the same for linear equations in more variables?
Definitions
A linear equation is an expression
a1 x1 + a2 x2 + · · · + an xn = b
where n ↓ 1, a1 , . . . , an are real numbers, not all of them equal to zero, and b is a real number.
A system of linear equations is a set of m ↓ 1 linear equations. It is not required that m = n.
A solution to a system of m equations in n variables is an n-tuple of numbers that satisfy each of the
equations.
Solve a system means ‘find all solutions to the system.’
1-1
, 1-2 Section 1.1: Solutions and Elementary Operations
Systems of Linear Equations
A system of linear equations:
x1 ↔ 2x2 ↔ 7x3 = ↔1
↔x1 + 3x2 + 6x3 = 0
• variables: x1 , x2 , x3 .
• coe!cients:
1x1 ↔ 2x2 ↔ 7x3 = ↔1
↔1x1 + 3x2 + 6x3 = 0
• constant terms:
x1 ↔ 2x2 ↔ 7x3 = ↔1
↔x1 + 3x2 + 6x3 = 0
x1 = ↔3, x2 = ↔1, x3 = 0 is a solution to the system 2 solutions
H
x1 ↔ 2x2 ↔ 7x3 = ↔1 the
fits into
↔x1 + 3x2 + 6x3 = 0
infinitely many
solution
category
because
(↔3) ↔ 2(↔1) ↔ 7·0 = ↔1
↔(↔3) + 3(↔1) + 6·0 = 0.
Another solution to the system is x1 = 6, x2 = 0, x3 = 1.
However, x1 = ↔1, x2 = 0, x3 = 0 is not a solution to the system, because
(↔1) ↔ 2·0 ↔ 7·0 = ↔1
↔(↔1) + 3·0 + 6·0 = 1 →= 0
The system above is consistent, meaning that the system has at least one solution.
x1 + x2 + x3 = 0
x1 + x2 + x3 = ↔8
is an example of an inconsistent system, meaning that it has no solutions.
NATHANIEL HALL
,UNIT ONE
, Mathematical Structure II Spring 2025
Section 1.1: Solutions and Elementary Operations
Lecturer: Mr. Hall Math 170B
Motivation
Find all solutions of the (linear) equation in one variable:
ax = b
Solution
• If a →= 0, there is a unique solution x = b/a.
• Else if a = 0 and
1) b →= 0, there is no solution. >
- contradiction
2) b = 0, there are infinitely many solutions, in fact any x ↑ R is a solution. stautology
This is a complete description of all possible solutions of ax = b.
Can we do the same for linear equations in more variables?
Definitions
A linear equation is an expression
a1 x1 + a2 x2 + · · · + an xn = b
where n ↓ 1, a1 , . . . , an are real numbers, not all of them equal to zero, and b is a real number.
A system of linear equations is a set of m ↓ 1 linear equations. It is not required that m = n.
A solution to a system of m equations in n variables is an n-tuple of numbers that satisfy each of the
equations.
Solve a system means ‘find all solutions to the system.’
1-1
, 1-2 Section 1.1: Solutions and Elementary Operations
Systems of Linear Equations
A system of linear equations:
x1 ↔ 2x2 ↔ 7x3 = ↔1
↔x1 + 3x2 + 6x3 = 0
• variables: x1 , x2 , x3 .
• coe!cients:
1x1 ↔ 2x2 ↔ 7x3 = ↔1
↔1x1 + 3x2 + 6x3 = 0
• constant terms:
x1 ↔ 2x2 ↔ 7x3 = ↔1
↔x1 + 3x2 + 6x3 = 0
x1 = ↔3, x2 = ↔1, x3 = 0 is a solution to the system 2 solutions
H
x1 ↔ 2x2 ↔ 7x3 = ↔1 the
fits into
↔x1 + 3x2 + 6x3 = 0
infinitely many
solution
category
because
(↔3) ↔ 2(↔1) ↔ 7·0 = ↔1
↔(↔3) + 3(↔1) + 6·0 = 0.
Another solution to the system is x1 = 6, x2 = 0, x3 = 1.
However, x1 = ↔1, x2 = 0, x3 = 0 is not a solution to the system, because
(↔1) ↔ 2·0 ↔ 7·0 = ↔1
↔(↔1) + 3·0 + 6·0 = 1 →= 0
The system above is consistent, meaning that the system has at least one solution.
x1 + x2 + x3 = 0
x1 + x2 + x3 = ↔8
is an example of an inconsistent system, meaning that it has no solutions.
