Week 1 (1/19-1/22) Worksheet 1
DIS 119/120 GSI Xiaohan Yan
1 Review
DEFINITIONS
• linear equation, linear system;
• coe�cient, R and C, variable;
• solution, solution set, descriptive form of solution sets;
• consistent/inconsistent linear systems, equivalent linear systems.
2 Problems
Example 1. Solve the following linear systems
(a) x = 3.
(b)
#x ` 2y “ 4
3x ` y “ 7.
(c)
#x ` 2y “ 3
3x ` 6y “ 4.
(d)
#2x ` y “ 4
x ` 3y “ 7.
(e)
$’&
’%
x1 ` x2 ` x3 “ 3
x1 ` 2x2 ` 3x3 “ ´1
x1 ` 3x2 ` 5x3 “ ´5
.
Example 2. Think about the following questions
• Among the linear systems in 1, which are consistent? Inconsistent? Arethere equivalent linear systems?
• Compare the solution sets in (b) and (d), what do you see, and why?
1
egx x xn b 130140250320
imaginary x e4 Xrationalnumbers a atbila.bep.IM a
bigca.brealnumbers unknowns r nD geioft
dafb.io xxcomplexnumbersvi i aarctanda realausolutions
valuesofvarious tanoka solutionsetofe.gl2o4asb.a.boa.btRsolutionsexist cosolution havethesamesolutionsete.g 6 8 103 40car
3 4 5320
x 3 33 uniquesolution
x 42y x zHy42,41 12in uniquesolution
zµzy y 7 I y i
µ y q a sow y ya
HaysIlia's 11h44uniquesolutionyet
3 97 y anyayx cb id
Example 3. Find the value of c such that the following system is inconsistent#
x1 ` cx2 “ ´1
2x1 ´ 2x2 “ 0.
Example 4. Find the value of the coe�cient c such that the following twosystems are equivalent
#x1 ´ cx2 “ 0
x1 ` x3 “ 0,
#2x1 ´ x2 ` x3 “ 0
x2 ` x3 “ 0.
2
OA B
Example3 Method inconsistent contradictionsamonglinearequations
x x oe's fifty
1 Itake j 2 2912 4
MethodI I C 1 reduceto RREF2 2 o
lastcolumnispivotsolves
Example4 solutionset ofBis fl ai a.at aelR
ofA is l a a all acIRIPREF equivalentsystemshavethesameKREF