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