Solving Linear Equations: Definitions and Steps
Linear equations of the form ax + b = cx + d are explored, along with steps to solve them. Definitions and examples of linear equations are provided, along with cases of no solution or many solutions. The content includes images illustrating the process of solving linear equations.
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Presentation Transcript
4.2.2 Solving equations of the form ax + b = cx + d 1
Definition Linear equation [an equation] that is equivalent to an equation of the form ?? + ? = ?? + ?, where either ? 0 or ? 0 2
0. ?? + ? = ?? + ? 1. ?? + ? ?? = ?? + ? ?? 2. ?? ?? + ? = ?? ?? + ? 3. ? ? ? + ? = 0 + ? = ? 4. ? ? ? + ? ? = ? ? 5. ? ? ? = ? ? Steps for Solving Linear Equations 3
0. ?? + ? = ?? + ? 1. ?? + ? ?? = ?? + ? ?? 2. ?? ?? + ? = ?? ?? + ? 3. ? ? ? + ? = 0 + ? = ? 4. ? ? ? + ? ? = ? ? 5. ? ? ? = ? ? Steps for Solving Linear Equations 4
0. ?? + ? = ?? + ? 1. ?? + ? ?? = ?? + ? ?? 2. ?? ?? + ? = ?? ?? + ? 3. ? ? ? + ? = 0 + ? = ? 4. ? ? ? + ? ? = ? ? 5. ? ? ? = ? ? Steps for Solving Linear Equations 5
0. ?? + ? = ?? + ? 1. ?? + ? ?? = ?? + ? ?? 2. ?? ?? + ? = ?? ?? + ? 3. ? ? ? + ? = 0 + ? = ? 4. ? ? ? + ? ? = ? ? 5. ? ? ? = ? ? Steps for Solving Linear Equations 6
0. ?? + ? = ?? + ? 1. ?? + ? ?? = ?? + ? ?? 2. ?? ?? + ? = ?? ?? + ? 3. ? ? ? + ? = 0 + ? = ? 4. ? ? ? + ? ? = ? ? 5. ? ? ? = ? ? Steps for Solving Linear Equations 7
0. ?? + ? = ?? + ? 1. ?? + ? ?? = ?? + ? ?? 2. ?? ?? + ? = ?? ?? + ? 3. ? ? ? + ? = 0 + ? = ? 4. ? ? ? + ? ? = ? ? 5. ? ? ? = ? ? Steps for Solving Linear Equations 8
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. 0? = ? i. ii. ? ? 0 no solution ? ? = 0 many solutions Case 1: ? ? = 0 9
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. 0? = ? i. ii. ? ? 0 no solution ? ? = 0 many solutions Case 1: ? ? = 0 10
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. 0? = ? i. ii. ? ? 0 no solution ? ? = 0 many solutions Case 1: ? ? = 0 11
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. ?? = ? such that ? 0 ?? = ?has a unique solution 1 ? = ? ? and B = (d b) Apply group properties ? =1 ? ? = ? ? Case 2: ? ? 0 ? ? ? ? =? ? 1 ? ? 12
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. ?? = ? such that ? 0 ?? = ?has a unique solution 1 ? = ? ? and B = (d b) Apply group properties ? =1 ? ? = ? ? Case 2: ? ? 0 ? ? ? ? =? ? 1 ? ? 13
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. ?? = ? such that ? 0 ?? = ?has a unique solution 1 ? = ? ? and B = (d b) Apply group properties ? =1 ? ? = ? ? Case 2: ? ? 0 ? ? ? ? =? ? 1 ? ? 14
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. ?? = ? such that ? 0 ?? = ?has a unique solution 1 ? = ? ? and B = (d b) Apply group properties ? =1 ? ? = ? ? Case 2: ? ? 0 ? ? ? ? =? ? 1 ? ? 15
Closure under multiplication Associative under multiplication Multiplicative Identity Multiplicative Inverse Group Properties 16
Steps for Solving Linear Equations ? ? ? = ? ? 5. 6. ?? = ? such that ? 0 ?? = ?has a unique solution 1 ? = ? ? and B = (d b) Apply group properties ? =1 ? ? = ? ? Case 2: ? ? 0 ? ? ? ? =? ? 1 ? ? 17
a set with two binary operations (+ and ) that satisfies the field properties (1) Closure for the two binary operations (2) Associativity of Multiplication (3) Associativity of Addition (4) Additive Identity (5) Multiplicative Identity (6) Commutativity of Addition (7) Commutativity of Multiplication (8) Additive Inverse (9) Multiplicative Inverse* (10)Distributivity of Multiplication over Addition Definitions field < ?,+, > field properties 18
If a, b, c, and d are in a set S, < ?,+,> is a field, and ? ?, then the equation ?? + ? = ?? + ? has exactly one solution, ? ? ? ?, in S. If ? = ?, then ?? + ? = ?? + ? has no Theorem 4.3: solution if ? ? and any element of S as a solution if ? = ?. Idea: All linear equations in a field can be solved without going outside of the field. 19
If a, b, c, and d are in a set S, < ?,+,> is a field, and ? ?, then the equation ?? + ? = ?? + ? has exactly one solution, ? ? ? ?, in S. If ? = ?, then ?? + ? = ?? + ? has no Theorem 4.3: solution if ? ? and any element of S as a solution if ? = ?. Idea: All linear equations in a field can be solved without going outside of the field. 20
Is the system < ,+, > of real numbers a field? 21
Field Properties Field Properties (1) Closure for the two binary operations (2) Associativity of Multiplication (3) Associativity of Addition (4) Additive Identity (5) Multiplicative Identity (6) Commutativity of Addition (7) Commutativity of Multiplication (8) Additive Inverse (9) Multiplicative Inverse* (10)Distributivity of Multiplication over Addition Some Some Properties Properties Closure Property of Addition Closure Property of Multiplication Associative Property of Multiplication Additive Identity Property Multiplicative Identity Property Commutative Property of Addition Commutative Property of Multiplication Additive Inverse Property Multiplicative Inverse Property Distributive Property 22
The system < ,+, > of real numbers is a field 23
Is system < ,+, > of complex numbers a field? 24
Field Properties Field Properties (1) Closure for the two binary operations (2) Associativity of Multiplication (3) Associativity of Addition (4) Additive Identity (5) Multiplicative Identity (6) Commutativity of Addition (7) Commutativity of Multiplication (8) Additive Inverse (9) Multiplicative Inverse* (10)Distributivity of Multiplication over Addition Properties Properties Closure Commutativity (+ and ) Associativity (+ and ) Identities (+ and ) Additive Inverse Multiplicative Inverse Distributive Property 25
The system of complex numbers < ,+, > is a field 26
Properties to check if a subset of or a subset of is a field 1. S is closed under addition and multiplication 2. S contains, 0, 1, opposites (identities), and inverses of its elements 27
Proof: ? ?+? ?=?? + ?? ?? ? ? ? ?=?? ?? Example: Claim: < ,+, > is a field 0 ? ? ? ? ? ?,? 0 ? ? ?+0 ?=?? + 0 =? ? ?? ? ? ? ?=?? ??=? ? 28
Proof: Let 2 = ? + ? 2 ?,? } ? + ? 2 + ? + ? 2 = ? + ? + ? + ? 2 Example: Claim: < [ 2],+, > is a field 2 ? + ? 2 ? + ? 2 = ?? + ?? 2 + ?? 2 + ?? 2 2 = ?? + 2?? + ?? + ?? 0 = 0 + 0 2 ? + ? 2 = ? + ( ?) 2 1 ? + ? 2= (? + ? 2)(? ? 2) =? ? 2 ?2 2?2 2 1 = 1 + 0 2 [ 2] 2 ? ? 2 ? 2 [ 2] ?2 2?2 29
Common Subsets of that do not form fields with addition and multiplication Integers Even Integers ???? Irrational Numbers ? All Prime Numbers 30
Example: Claim: < ????,+, > is not a field Consider the linear equation 4? + 6 = 12? + 14 All the coefficients are in ????, but there is no even solution to this linear equation. 31
Systems of Equations ? ?= [? {?? + ?? = ? ?? + ?? = ? ? ? ? ?] ? 32
?1? ?2? ??? ?11 ?21 ??1 ?12 ?22 ??2 Definitions matrix A rectangular array dimension (A matrix with m rows and n column) ? ? 33
Definition Equal Matrices (two matrices are equal if and only if) each of the mn pairs of corresponding elements are equal 34
?1? ?2? ??? ?11 ?21 ??1 ?12 ?22 ??2 ? = Matrix Multiplication ?1? ?2? ??? ?11 ?21 ?n1 ?12 ?22 ??2 ? = 35
?1? ?2? ??? ?11 ?21 ??1 ?12 ?22 ??2 ? = Matrix Multiplication ?1? ?2? ??? ?11 ?21 ?n1 ?12 ?22 ??2 ? = 36
? ?= [? {?? + ?? = ? ?? + ?? = ? ? ? ? Matrix Multiplication ?] ? 37
{?? + ?? = ? ?? + ?? = ? ? ?= Systems of Equations ? ? ?? = ? ? ? ? ? 38
Is the set of 2 2 matrices ? group with ? ? ?a ? 39
With multiplication, the set of 2 2 matrices ? ? group, and every equation of the form ? ? ? ? ? ? with ?? ?? 0 is a Theorem 4.4: ? ? has a unique solution ? ? = 40
? ? ? ? =? ? ?,? = ? Closure under Multiplication ?? + ?? ?? + ?? ?? + ? ?? + ? ?? = 41
Associative under multiplication Section 4.2.2 Homework Problem #6 Prove that multiplication of 2 2 matrices is associative 42
Exists if AX = A. i.e. ? ? ? ? ? ? ?=? ? ? ? ? ?? + ?? ?? + ?? ?? + ?? ?? + ?? =? ? ? Multiplicative Identity ? The systems of equations {?? + ?? = ? ?? + ?? = ? and {?? + ?? = ? ?? + ?? = ? 43
Multiplicative Identity the multiplicative identity of a 2 2 matrix is 1 0 0 1 44
To find the multiplicative inverse of ? =? ? equation ? ? ? ? ? ?? + ?? = 1 ? ?, we need to solve the matrix =1 0 1 ? ? ? 0 ?? + ?? = 0 and {?? + ?? = 0 Section 4.2.2 Homework Problem #7a Show steps in solving the systems {?? + ?? = 1 Multiplicative Inverse ?? + ?? = 1 ?? + ?? = 0 and {?? + ?? = 0 For w, x, y, and z in terms of a, b, c, and d without using matrices. ?? + ?? = 1 45
To find the multiplicative inverse of ? =? ? equation ? ? ? ? ? {?? + ?? = 1 ? ?, we need to solve the matrix =1 0 1 ? ? ? 0 ?? + ?? = 0 and {?? + ?? = 0 Section 4.2.2 Homework Problem #7a Show steps in solving the systems {?? + ?? = 1 Multiplicative Inverse ?? + ?? = 1 ?? + ?? = 0 and {?? + ?? = 0 For w, x, y, and z in terms of a, b, c, and d without using matrices. ?? + ?? = 1 46
To find the multiplicative inverse of ? =? ? equation ? ? ? ? ? {?? + ?? = 1 ? ?, we need to solve the matrix =1 0 1 ? ? ? 0 ?? + ?? = 0 and {?? + ?? = 0 Section 4.2.2 Homework Problem #7a Show steps in solving the systems {?? + ?? = 1 Multiplicative Inverse ?? + ?? = 1 ?? + ?? = 0 and {?? + ?? = 0 For w, x, y, and z in terms of a, b, c, and d without using matrices. ?? + ?? = 1 47
Section 4.2.2 Homework Problem #7a the multiplicative inverse of a 2 2 matrix Multiplicative Inverse ? ? ? ? ? ?= ?? ?? ? ?? ?? ?? ?? ? ?? ?? provided that ?? ?? 0 48
The set of 2 2 matrices ? where the determinant 0 is a group. ? ? ? 49
? ?= ? ? ? ? ? ? ? ?, X = ? ?? = ? ?, and B = ? Let ? =? ? ? 2 2 Matrix Equation Solution Case 1: ?? ?? 0 ? 1 exists ? 1? ? = ? 1 ? ? 1 ? ? = ? 1 ? ? ? = ? 1 ? ? = ? 1 ? the lines ?? + ?? = ? and ?? + ?? = ? are not parallel and they intersect at one point 50
