4.3 cramer’s rule
•Download as PPT, PDF•
2 likes•3,927 views
The document provides information about solving systems of linear equations using Cramer's rule, including: - Cramer's rule uses determinants of the coefficient matrix to solve systems of linear equations. - For a 2x2 system, if the determinant of the coefficient matrix is not equal to 0, there is exactly one solution. - The solutions are found by taking the determinants of the numerator matrices (with one column replaced by the constants) divided by the determinant of the coefficient matrix. - Several examples demonstrate solving 2x2 systems using Cramer's rule.
Report
Share
Report
Share
![09.16.10/09.17.10 ACT OPENER ,[object Object],[object Object],[object Object],[object Object],[object Object],[object Object],3. Solve the system of linear equations by graphing. x + 2y = -4 4y = 3x + 12 ,[object Object],[object Object],[object Object],[object Object],[object Object]](https://image.slidesharecdn.com/4-3cramersrule-100916183522-phpapp01/85/4-3-cramer-s-rule-1-320.jpg)
![[object Object],[object Object]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
Recommended
systems of linear equations & matrices
This document provides an overview of matrices and matrix operations. It defines what a matrix is and discusses matrix order and elements. It then covers basic matrix operations like scalar multiplication, addition, and multiplication. It introduces the concepts of transpose, special matrices like diagonal and triangular matrices, and the null and identity matrices. The document aims to define fundamental matrix concepts and arithmetic operations.
CRAMER’S RULE
Kishan Kasundra presented on Cramer's rule for solving systems of equations. Cramer's rule uses determinants of coefficient matrices to find variables. It can be applied to systems with 2 or 3 equations. For 2 equations, the determinant of the original coefficient matrix and matrices with the coefficient columns replaced by the constants are calculated. The ratios of these determinants over the original give the variables. For 3 equations, a similar process is followed using a 3x3 coefficient matrix. An example showed applying Cramer's rule to solve a 2x2 system.
Lesson 5: Matrix Algebra (slides)
This document provides an overview of matrix algebra concepts including:
- Matrix addition is defined as adding corresponding elements and is commutative and associative.
- Matrix multiplication is defined as taking the dot product of rows and columns. It is associative but not commutative.
- The transpose of a matrix is obtained by flipping rows and columns.
- Properties of matrix operations like addition, multiplication, and transposition are discussed.
Linear Algebra
This document provides an overview and review of key concepts in linear algebra for a course. It defines matrices and basic matrix operations such as addition, subtraction and multiplication. It discusses vector operations including dot products, cross products and magnitudes. It also covers determinants, inverses, homogeneous coordinates and orthonormal bases. Resources for further information are provided.
Cramers rule
This document provides an overview of determinants and Cramer's rule for solving systems of linear equations. It begins with examples of calculating the determinants of 2x2 and 3x3 matrices. It then explains how to use Cramer's rule to determine if a system has a unique solution, no solution, or infinitely many solutions. The document concludes with examples applying Cramer's rule and determinants to solve word problems involving systems of linear equations with two or three variables.
Determinants. Cramer’s Rule
This document defines determinants and provides examples of calculating determinants of 2x2 and 3x3 matrices. It introduces key terminology like minors, cofactors, and expands on Cramer's rule for solving systems of linear equations using determinants. The document explains that the determinant of a square matrix is a single number associated with the matrix, and provides rules and examples for calculating determinants by summing the products of elements and their cofactors.
System Of Linear Equations
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
Linear Algebra and Matrix
The document provides an overview of linear algebra and matrix theory. It discusses the history and development of matrices, defines key matrix concepts like dimensions and operations, and covers foundational topics like matrix addition, multiplication, inverses, and solving systems of linear equations. The document is intended as an introduction to linear algebra and matrices for students.
Recommended
systems of linear equations & matrices
This document provides an overview of matrices and matrix operations. It defines what a matrix is and discusses matrix order and elements. It then covers basic matrix operations like scalar multiplication, addition, and multiplication. It introduces the concepts of transpose, special matrices like diagonal and triangular matrices, and the null and identity matrices. The document aims to define fundamental matrix concepts and arithmetic operations.
CRAMER’S RULE
Kishan Kasundra presented on Cramer's rule for solving systems of equations. Cramer's rule uses determinants of coefficient matrices to find variables. It can be applied to systems with 2 or 3 equations. For 2 equations, the determinant of the original coefficient matrix and matrices with the coefficient columns replaced by the constants are calculated. The ratios of these determinants over the original give the variables. For 3 equations, a similar process is followed using a 3x3 coefficient matrix. An example showed applying Cramer's rule to solve a 2x2 system.
Lesson 5: Matrix Algebra (slides)
This document provides an overview of matrix algebra concepts including:
- Matrix addition is defined as adding corresponding elements and is commutative and associative.
- Matrix multiplication is defined as taking the dot product of rows and columns. It is associative but not commutative.
- The transpose of a matrix is obtained by flipping rows and columns.
- Properties of matrix operations like addition, multiplication, and transposition are discussed.
Linear Algebra
This document provides an overview and review of key concepts in linear algebra for a course. It defines matrices and basic matrix operations such as addition, subtraction and multiplication. It discusses vector operations including dot products, cross products and magnitudes. It also covers determinants, inverses, homogeneous coordinates and orthonormal bases. Resources for further information are provided.
Cramers rule
This document provides an overview of determinants and Cramer's rule for solving systems of linear equations. It begins with examples of calculating the determinants of 2x2 and 3x3 matrices. It then explains how to use Cramer's rule to determine if a system has a unique solution, no solution, or infinitely many solutions. The document concludes with examples applying Cramer's rule and determinants to solve word problems involving systems of linear equations with two or three variables.
Determinants. Cramer’s Rule
This document defines determinants and provides examples of calculating determinants of 2x2 and 3x3 matrices. It introduces key terminology like minors, cofactors, and expands on Cramer's rule for solving systems of linear equations using determinants. The document explains that the determinant of a square matrix is a single number associated with the matrix, and provides rules and examples for calculating determinants by summing the products of elements and their cofactors.
System Of Linear Equations
For a system involving two variables (x and y), each linear equation determines a line on the xy-plane. Because a solution to a linear system must satisfy all of the equations, the solution set is the intersection of these lines, and is hence either a line, a single point, or the empty set
Linear Algebra and Matrix
The document provides an overview of linear algebra and matrix theory. It discusses the history and development of matrices, defines key matrix concepts like dimensions and operations, and covers foundational topics like matrix addition, multiplication, inverses, and solving systems of linear equations. The document is intended as an introduction to linear algebra and matrices for students.
Quadratic Equation
A quick intro. to Quadratic Equation.
Few examples have also been discussed to make the student grasp the concept easily.
Matrices and determinants assignment
This document provides a mathematics assignment on matrices and determinants containing 25 problems of varying difficulty levels. The problems involve calculating determinants, finding cofactors, inverses of matrices, solving systems of equations using matrices, expressing matrices as sums, and proving identities about determinants. Students are asked to perform operations on matrices, solve equations involving matrices, use properties of determinants, and apply concepts of invertibility and elementary row operations.
Tutorial linear equations and linear inequalities
This document discusses linear equations and inequalities in one variable. It begins by defining open sentences, variables, and solutions. It then covers topics like solving linear equations using addition, subtraction, multiplication, and division. It also discusses solving multi-step equations. Graphing solutions to equations is explained. The document also covers understanding and solving linear inequalities in one variable as well as graphing inequalities. It provides examples of how equations and inequalities can be applied to everyday situations.
Project business maths
The history and development of matrix theory is summarized as follows:
1) The term "matrix" was introduced in 1850 by James Joseph Sylvester to describe rectangular arrays of numbers or expressions arranged in rows and columns.
2) The founder of modern matrix theory is considered to be Arthur Cayley, who in the 1850s introduced concepts such as inverse matrices and matrix multiplication.
3) Important developments in matrix theory continued throughout the 19th and 20th centuries, including the discovery by Arthur Cayley and William Hamilton of unique properties of matrices.
Cramer's Rule
1) Cramer's rule can be used to solve systems of linear equations. It expresses the solution in terms of the determinants of the coefficient matrix and matrices with one column replaced by the constants vector.
2) If the determinant of the coefficient matrix is non-zero, there is a unique solution. If it is zero, there may be no solution or infinitely many solutions.
3) Three examples demonstrate applying Cramer's rule to find the unique solution, that there is no solution, and that there are infinitely many solutions, respectively.
QUADRATIC EQUATIONS
This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.
Co-factor matrix..
The document describes the cofactor method for finding the inverse of a matrix A. It defines the cofactor Cij as the signed determinant of the matrix made by removing row i and column j from A. The inverse is then given by the transpose of the matrix of cofactors divided by the determinant of A. An example calculates the inverse of the 2x2 matrix A = [a, c; b, d] using this method.
Matrix Algebra seminar ppt
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
system linear equations and matrices
row echlon , reduced row echlon, hermitian, skew hermitian matrices etc. according to GTU SYLLABUS (VCLA)
Complex number
The document provides an introduction to complex numbers including:
- Combining real and imaginary numbers like 4 - 3i.
- Properties of i including i2 = -1.
- Converting complex numbers between Cartesian, polar, trigonometric and exponential forms.
- Operations on complex numbers such as addition, subtraction, multiplication and division.
- Comparing real and imaginary parts of complex numbers when solving equations.
Mathematics
Mathematics is the study of quantity, structure, space, and change. It originated from practical needs to count and measure, and early forms can be seen in patterns on ancient structures. Important historical figures like Pythagoras, Pascal, and Euler made significant contributions to fields like geometry, calculus, and algebra. Mathematics is divided into pure mathematics, which explores logic and reasoning, and applied mathematics, which uses mathematical methods in fields like science, engineering, and business. Common branches of mathematics include arithmetic, algebra, geometry, sets, probability, and calculus.
Pair of linear equation in two variable
This document discusses linear equations in two variables. It defines a linear equation as an equation between two variables that forms a straight line when graphed. It then defines a linear equation in two variables as an equation with two variables, usually x and y, where the variables are multiplied by a number or added to another term. The document goes on to explain that a system of linear equations can have one solution, no solution, or infinitely many solutions depending on whether the lines intersect at one point, do not intersect, or coincide. It describes algebraic and graphical methods for solving systems of linear equations, focusing on substitution, elimination, and cross-multiplication algebraic methods.
MATRICES
This document defines and describes different types of matrices including:
- Upper and lower triangular matrices
- Determinants which are scalars obtained from products of matrix elements according to constraints
- Band matrices which are sparse matrices with nonzero elements confined to diagonals
- Transpose matrices which exchange the rows and columns of a matrix
- Inverse matrices which when multiplied by the original matrix produce the identity matrix
Linear equations in 2 variables
This document discusses linear equations. It defines linear equations as algebraic equations with terms that are constants or the product of constants and variables. Linear equations can have one or more variables. The document describes variables, constants, and examples of linear equations with one and two variables. It explains how to graph and solve systems of linear equations using graphical and algebraic methods like elimination and cross multiplication. Graphical methods involve plotting the lines defined by each equation and finding their point(s) of intersection. Algebraic methods eliminate variables to solve for the remaining ones.
AS LEVEL QUADRATIC (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS
This document discusses completing the square to write quadratic equations in standard form. It explains how to write quadratics in the form (x - h)2 + k when the leading coefficient a is positive, negative, or not equal to 1. The standard form allows easily finding the vertex (h, k) and sketching the graph. It also covers using the discriminant to determine the number of roots, solving quadratic inequalities, and relating the sign of a to whether the graph opens up or down.
Matrices ppt
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
Determinants - Mathematics
The document discusses determinants and their properties. It defines determinants as scalars associated with square matrices. Determinants can be computed for matrices of any size using expansions by cofactors along rows or columns. The key properties of determinants are that they are unchanged by row operations and that the inverse of a non-singular matrix can be computed using the adjugate matrix divided by the determinant. The document provides examples of computing the determinant, adjugate, and inverse of matrices.
Upcat math 2014
This document contains 54 multiple choice questions testing various math and problem solving skills. The questions cover topics like arithmetic, algebra, geometry, ratios, functions and more. They range in difficulty from basic calculations to more complex word problems requiring multiple steps to solve. The full set of questions and possible answer choices are provided for review.
PPT on Linear Equations in two variables
This ppt on different methods of solving equations like substitution method, elimination method, and cross multiplication method.
Absolute Value Functions & Graphs
Absolute value functions have a V-shaped graph. There are three main ways to graph an absolute value function: using a table of values, using a graphing calculator, or interpreting the equation. The vertex of an absolute value graph is always the x-coordinate inside the absolute value signs. To graph, one finds the vertex and then plots additional points on either side, connecting them to form the V-shape. The number outside the absolute value signs moves the graph up or down, while the number inside moves it left or right.
4.3 Determinants and Cramer's Rule
This document discusses determinants and Cramer's rule. It defines determinants as special numbers associated with square matrices. It provides examples of calculating determinants of 2x2 and 3x3 matrices by rewriting columns and multiplying diagonals. The document also introduces Cramer's rule as a method to solve linear systems using determinants, where the variable is replaced with constants in the coefficient matrix. Examples are given to demonstrate solving 2x2 and 3x3 systems using Cramer's rule.
Cramer's Rule System of Equations
The document describes using Cramer's rule to solve a system of two equations with two unknowns (5x + 2y = 16, 3x - 5y = -9). It shows setting up the equations in a matrix and calculating the determinants to find that the solutions are x=2 and y=3.
More Related Content
What's hot
Quadratic Equation
A quick intro. to Quadratic Equation.
Few examples have also been discussed to make the student grasp the concept easily.
Matrices and determinants assignment
This document provides a mathematics assignment on matrices and determinants containing 25 problems of varying difficulty levels. The problems involve calculating determinants, finding cofactors, inverses of matrices, solving systems of equations using matrices, expressing matrices as sums, and proving identities about determinants. Students are asked to perform operations on matrices, solve equations involving matrices, use properties of determinants, and apply concepts of invertibility and elementary row operations.
Tutorial linear equations and linear inequalities
This document discusses linear equations and inequalities in one variable. It begins by defining open sentences, variables, and solutions. It then covers topics like solving linear equations using addition, subtraction, multiplication, and division. It also discusses solving multi-step equations. Graphing solutions to equations is explained. The document also covers understanding and solving linear inequalities in one variable as well as graphing inequalities. It provides examples of how equations and inequalities can be applied to everyday situations.
Project business maths
The history and development of matrix theory is summarized as follows:
1) The term "matrix" was introduced in 1850 by James Joseph Sylvester to describe rectangular arrays of numbers or expressions arranged in rows and columns.
2) The founder of modern matrix theory is considered to be Arthur Cayley, who in the 1850s introduced concepts such as inverse matrices and matrix multiplication.
3) Important developments in matrix theory continued throughout the 19th and 20th centuries, including the discovery by Arthur Cayley and William Hamilton of unique properties of matrices.
Cramer's Rule
1) Cramer's rule can be used to solve systems of linear equations. It expresses the solution in terms of the determinants of the coefficient matrix and matrices with one column replaced by the constants vector.
2) If the determinant of the coefficient matrix is non-zero, there is a unique solution. If it is zero, there may be no solution or infinitely many solutions.
3) Three examples demonstrate applying Cramer's rule to find the unique solution, that there is no solution, and that there are infinitely many solutions, respectively.
QUADRATIC EQUATIONS
This document discusses four methods for solving quadratic equations: factorization, completing the square, using a formula, and using graphs. It provides an example of solving the equation 2x^2 - 10x + 12 + x^2 + 6x = -9 by factorizing into (x - 3)(x - 1) = 0, finding that the solutions are x = 3 or x = 1.
Co-factor matrix..
The document describes the cofactor method for finding the inverse of a matrix A. It defines the cofactor Cij as the signed determinant of the matrix made by removing row i and column j from A. The inverse is then given by the transpose of the matrix of cofactors divided by the determinant of A. An example calculates the inverse of the 2x2 matrix A = [a, c; b, d] using this method.
Matrix Algebra seminar ppt
It contains the basics of matrix which includes matrix definition,types of matrices,operations on matrices,transpose of matrix,symmetric and skew symmetric matrix,invertible matrix,
application of matrix.
system linear equations and matrices
row echlon , reduced row echlon, hermitian, skew hermitian matrices etc. according to GTU SYLLABUS (VCLA)
Complex number
The document provides an introduction to complex numbers including:
- Combining real and imaginary numbers like 4 - 3i.
- Properties of i including i2 = -1.
- Converting complex numbers between Cartesian, polar, trigonometric and exponential forms.
- Operations on complex numbers such as addition, subtraction, multiplication and division.
- Comparing real and imaginary parts of complex numbers when solving equations.
Mathematics
Mathematics is the study of quantity, structure, space, and change. It originated from practical needs to count and measure, and early forms can be seen in patterns on ancient structures. Important historical figures like Pythagoras, Pascal, and Euler made significant contributions to fields like geometry, calculus, and algebra. Mathematics is divided into pure mathematics, which explores logic and reasoning, and applied mathematics, which uses mathematical methods in fields like science, engineering, and business. Common branches of mathematics include arithmetic, algebra, geometry, sets, probability, and calculus.
Pair of linear equation in two variable
This document discusses linear equations in two variables. It defines a linear equation as an equation between two variables that forms a straight line when graphed. It then defines a linear equation in two variables as an equation with two variables, usually x and y, where the variables are multiplied by a number or added to another term. The document goes on to explain that a system of linear equations can have one solution, no solution, or infinitely many solutions depending on whether the lines intersect at one point, do not intersect, or coincide. It describes algebraic and graphical methods for solving systems of linear equations, focusing on substitution, elimination, and cross-multiplication algebraic methods.
MATRICES
This document defines and describes different types of matrices including:
- Upper and lower triangular matrices
- Determinants which are scalars obtained from products of matrix elements according to constraints
- Band matrices which are sparse matrices with nonzero elements confined to diagonals
- Transpose matrices which exchange the rows and columns of a matrix
- Inverse matrices which when multiplied by the original matrix produce the identity matrix
Linear equations in 2 variables
This document discusses linear equations. It defines linear equations as algebraic equations with terms that are constants or the product of constants and variables. Linear equations can have one or more variables. The document describes variables, constants, and examples of linear equations with one and two variables. It explains how to graph and solve systems of linear equations using graphical and algebraic methods like elimination and cross multiplication. Graphical methods involve plotting the lines defined by each equation and finding their point(s) of intersection. Algebraic methods eliminate variables to solve for the remaining ones.
AS LEVEL QUADRATIC (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS
This document discusses completing the square to write quadratic equations in standard form. It explains how to write quadratics in the form (x - h)2 + k when the leading coefficient a is positive, negative, or not equal to 1. The standard form allows easily finding the vertex (h, k) and sketching the graph. It also covers using the discriminant to determine the number of roots, solving quadratic inequalities, and relating the sign of a to whether the graph opens up or down.
Matrices ppt
Here are the key steps to find the eigenvalues of the given matrix:
1) Write the characteristic equation: det(A - λI) = 0
2) Expand the determinant: (1-λ)(-2-λ) - 4 = 0
3) Simplify and factor: λ(λ + 1)(λ + 2) = 0
4) Find the roots: λ1 = 0, λ2 = -1, λ3 = -2
Therefore, the eigenvalues of the given matrix are -1 and -2.
Determinants - Mathematics
The document discusses determinants and their properties. It defines determinants as scalars associated with square matrices. Determinants can be computed for matrices of any size using expansions by cofactors along rows or columns. The key properties of determinants are that they are unchanged by row operations and that the inverse of a non-singular matrix can be computed using the adjugate matrix divided by the determinant. The document provides examples of computing the determinant, adjugate, and inverse of matrices.
Upcat math 2014
This document contains 54 multiple choice questions testing various math and problem solving skills. The questions cover topics like arithmetic, algebra, geometry, ratios, functions and more. They range in difficulty from basic calculations to more complex word problems requiring multiple steps to solve. The full set of questions and possible answer choices are provided for review.
PPT on Linear Equations in two variables
This ppt on different methods of solving equations like substitution method, elimination method, and cross multiplication method.
Absolute Value Functions & Graphs
Absolute value functions have a V-shaped graph. There are three main ways to graph an absolute value function: using a table of values, using a graphing calculator, or interpreting the equation. The vertex of an absolute value graph is always the x-coordinate inside the absolute value signs. To graph, one finds the vertex and then plots additional points on either side, connecting them to form the V-shape. The number outside the absolute value signs moves the graph up or down, while the number inside moves it left or right.
What's hot (20)
AS LEVEL QUADRATIC (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS
AS LEVEL QUADRATIC (CIE) EXPLAINED WITH EXAMPLE AND DIAGRAMS
Viewers also liked
4.3 Determinants and Cramer's Rule
This document discusses determinants and Cramer's rule. It defines determinants as special numbers associated with square matrices. It provides examples of calculating determinants of 2x2 and 3x3 matrices by rewriting columns and multiplying diagonals. The document also introduces Cramer's rule as a method to solve linear systems using determinants, where the variable is replaced with constants in the coefficient matrix. Examples are given to demonstrate solving 2x2 and 3x3 systems using Cramer's rule.
Cramer's Rule System of Equations
The document describes using Cramer's rule to solve a system of two equations with two unknowns (5x + 2y = 16, 3x - 5y = -9). It shows setting up the equations in a matrix and calculating the determinants to find that the solutions are x=2 and y=3.
RANGKAIAN THEVENIN-NORTHON
Dokumen tersebut memberikan daftar nama dan merupakan bagian dari mata kuliah Fisika di salah satu perguruan tinggi. Tidak terdapat informasi lain yang relevan untuk diringkas.
Matrices - Cramer's Rule
1) The inverse of the matrix [[1, 2], [3, 4]] is [[0.25, -0.5], [-0.75, 1]].
2) The inverse of the matrix [[5, -1], [0, 3]] is [[0.2, 0.0333], [0, 0.3333]].
3) The inverse of the matrix [[1, 2, 0], [0, 3, 0], [0, 0, 5]] is [[1, -0.6667, 0], [0, 0.3333, 0], [0, 0, 0.
Ppt cramer rules
Dokumen tersebut membahaskan metode baru untuk menyelesaikan sistem persamaan linier bernama Petua Cramer. Petua Cramer hanya memerlukan penentu matriks untuk menentukan nilai satu variabel pada satu waktu, berbeda dengan metode sebelumnya yang memerlukan perhitungan banyak kofaktor yang rumit. Contoh penyelesaian sistem persamaan 3x3 menggunakan Petua Cramer kemudian diberikan.
Teori thevenin
Teori Thevenin menyatakan bahwa rangkaian listrik kompleks dapat disederhanakan menjadi sumber tegangan dan resistor tunggal yang setara. Teori ini berguna untuk menganalisis sistem daya dan rangkaian lainnya dengan mudah dengan mengubah rangkaian asli menjadi rangkaian ekuivalen Thevenin yang lebih sederhana. Penggunaan utama teori ini adalah untuk mempermudah perhitungan arus dan tegangan pada rangkaian.
Teorema Thevenin
Presentasi ini berupa penjelasan tentang teorema Thevenin yang disarikan dari video tutorial Prof.Dr. C.B. Bangal (Youtube) dan beberapa referensi lainnya. Teorema Thevenin merupakan salah satu teorema dalam analisis rangkaian elektronik. Terkadang mahasiswa mengalami kesulitan dalam memahami teorema ini. Prof. Dr. C.B. Bangal mengajak kita untuk memahami teorema ini langsung dengan penerapan soal. Keterangan dari Prof. Dr. C.B. Bangal tersebut kami ubah dalam bentuk slide bahasa Indonesia. Semoga dapat bermanfaat bagi para mahasiswa/pelajar yang sedang belajar dasar-dasar analisis rangkaian elektronika. Selamat belajar dan semoga sukses!
Principle of electricity
This document provides an overview of electrical circuits and key concepts like Ohm's Law. It discusses the basics of series and parallel circuits, and how to measure current and voltage. Key points covered include: Ohm's Law states that current is directly proportional to voltage and inversely proportional to resistance. Circuits can be connected in series, where components are in a row, or parallel, where there are multiple paths. Current is measured using an ammeter in series, and voltage is measured using a voltmeter connected across components. The document concludes with a restatement of Ohm's Law and definitions of series and parallel circuits.
MSc Research - Chin Seng Fatt
The document summarizes the design and fabrication of a CMOS ISFET for pH measurement. It describes the objectives of the research which are to simulate the ISFET using TCAD, design masks for fabrication, fabricate the ISFET using CMOS compatible materials and processes, and characterize the fabricated device. Key steps included TCAD simulation of the process and device, mask design involving 6 masks, and fabrication using a silicon wafer and processes like oxidation, diffusion, etching, and metallization.
Modul 4 WORKPLACE CULTURE ETHICS
Modul 4 membahas tentang budaya kerja dan etika di tempat kerja. Modul ini menjelaskan pentingnya integriti, tanggung jawab, dan penilaian prestasi yang adil bagi pekerja untuk menciptakan lingkungan kerja yang harmonis dan produktif.
TEOREM THEVENIN & TEOREM NORTON
INTRODUCTION TO THEVENIN THEOREM AND NORTON THEOREM AND ITS METHOD OF APPLICATIONS IN CIRCUIT ANALYSIS AND SIMPLIFICATION
Hukum Kirchoff / Kirchoff Law
Dokumen ini membahas Hukum Kirchoff untuk litar listrik, termasuk Hukum Kirchoff Pertama tentang arus dan Hukum Kirchoff Kedua tentang voltan. Dokumen ini menjelaskan definisi dan rumus untuk kedua hukum, memberikan contoh penyelesaian masalah litar menggunakan hukum-hukum tersebut, dan membandingkan penggunaan Hukum Kirchoff Arus dan Hukum Kirchoff Voltan.
Asas Elektrik
1. Arus elektrik ialah aliran elektron bebas dalam litar.
2. Struktur asas atom terdiri daripada nukleus yang mengandungi proton dan neutron, manakala elektron membentuk orbit sekeliling nukleus.
3. Pengalir membenarkan aliran elektron bebas, manakala penebat menghalang aliran elektron. Semikonduktor mempunyai jumlah elektron antara pengalir dan penebat.
hukum asas litar elektrik
Menerangkan jenis hukum dan formula asas yang digunakan dalam pengiraan voltan, rintangan dan arus suatu litar elektrik. Mnegandungi definisi hukum, formula matematik, contoh soalan dan contoh penyelesaian.
Mathematics
Mathematics is the systematic treatment of magnitude, relationships between figures and forms, and relations between quantities expressed symbolically. Some key types of math discussed include:
- Algebra - the study of operations and relations and the constructions arising from them. An example algebra equation is shown.
- Geometry - the study of shape, size, relative position of figures, and properties of space.
- Trigonometry - the computational component of geometry concerned with calculating unknown sides and angles of triangles.
- Calculus - focused on limits, functions, derivatives, integrals, and infinite series. It has two major branches: differential and integral calculus. Calculus has widespread applications and can solve problems algebra cannot.
Viewers also liked (17)
Similar to 4.3 cramer’s rule
Linear Systems
This document discusses linear systems of equations. It defines a linear system as one where the equations are of the first degree. It describes four methods for solving linear systems: substitution, comparison, reduction, and Cramer's method. It also discusses literal systems that contain parameters and fractional systems where variables appear in denominators. An example demonstrates solving a 3x3 system using substitution to find the unique solution.
Solving systems by substitution
Here are the steps to solve this problem:
Let x = one number
Let y = the other number
x + y = 84
x = 3y
x + 3y = 84
4y = 84
y = 21
x = 3(21) = 63
The numbers are 21 and 63.
Q1-W1-Factoring Polynomials.pptx
This document provides instruction on factoring polynomials. It covers several factoring methods including common monomial factoring, difference of squares, sum and difference of cubes, perfect square trinomials, and general trinomials. Examples are provided for each method. The objectives are to determine appropriate factoring methods, factor polynomials completely using various techniques, and solve problems involving polynomial factors.
This quiz is open book and open notes/tutorialoutlet
FOR MORE CLASSES VISIT
tutorialoutletdotcom
Math 107 Quiz 2 Spring 2017 OL4
Professor: Dr. Katiraie Name________________________________ Instructions: The quiz is worth 100 points. There are 10 problems, each worth 10 points. Your score
will be posted in your Portfolio with comments.
7.3 Solving Multi Step Frac Deci
This document provides an overview of solving multi-step equations with fractions and decimals. It discusses identifying coefficients, multiplying terms by the reciprocal of fractions to clear them, and two methods for working with decimal coefficients - either keeping them as decimals or clearing them by multiplying all terms by a power of 10. Sample problems are provided and worked through as examples.
Limits
The document discusses limits and limit theorems. It begins with an introduction to limits and how they are represented symbolically. It then presents several limit theorems including: 1) the identity function, where the limit of x as it approaches c is equal to c; 2) the constant rule, where the limit of a constant is itself; 3) the sum and difference rules, where the limit of a sum or difference is the sum or difference of the individual limits. The document demonstrates these theorems with examples and explains how they allow limits to be determined without a table or graph.
Daa chapter 3
Dynamic programming (DP) is a powerful technique for solving optimization problems by breaking them down into overlapping subproblems and storing the results of already solved subproblems. The document provides examples of how DP can be applied to problems like rod cutting, matrix chain multiplication, and longest common subsequence. It explains the key elements of DP, including optimal substructure (subproblems can be solved independently and combined to solve the overall problem) and overlapping subproblems (subproblems are solved repeatedly).
Ma3bfet par 10.7 7 aug 2014
The document contains information about partial fraction decomposition:
1. It discusses four cases for partial fraction decomposition based on the factors of the denominator: distinct linear factors, repeated linear factors, distinct irreducible quadratic factors, and repeated irreducible quadratic factors.
2. It provides examples to illustrate each case, showing how to set up and solve systems of equations to determine the coefficients of the partial fractions.
3. Homework Task 4 on systems of equations and inequalities is due on August 13, and consultation times for Ms. Durandt are on Thursdays and Fridays at 10:30.
Chapter 10
The document provides examples for solving one-step, two-step, and multi-step equations. It begins with warm-up problems and then works through examples of solving equations with variables on both sides, combining like terms, and clearing fractions by multiplying both sides by the least common denominator. The examples are accompanied by step-by-step explanations and checks of the solutions. Additional practice problems and lessons on solving equations are also included.
3 2 solving systems of equations (elimination method)
The document describes the elimination method for solving systems of equations. The key steps are:
1) Write both equations in standard form Ax + By = C
2) Determine which variable to eliminate using addition or subtraction
3) Solve the resulting equation for one variable
4) Substitute back into the original equation to solve for the other variable
5) Check that the solution satisfies both original equations
It provides examples showing how to set up and solve systems of equations using elimination, including word problems about supplementary angles and finding two numbers based on their sum and difference.
Simultaneous Equations Practical Construction
The document discusses solving simultaneous equations using algebraic methods and graphing. It provides examples of setting up and solving systems of two equations with two unknowns to find the values of the unknowns. Various word problems are presented and worked through step-by-step to show how to set up the appropriate equations to find the unknown values being asked about, such as costs, numbers of items, etc. Strategies for setting up simultaneous equations from word problems are emphasized.
Ca 1.6
This document provides an overview of quadratic equations, including definitions, methods for solving quadratic equations such as factoring, completing the square, and using the quadratic formula, and applications of quadratic equations. Key topics covered include defining linear and quadratic equations, solving quadratics by factoring when possible and using completing the square or the quadratic formula when not factorable, deriving the quadratic formula, interpreting the discriminant, and modeling real-world situations with quadratic equations.
April 22, 2015
The document discusses solving quadratic equations using the quadratic formula. It introduces the quadratic formula as -b ± √(b2 - 4ac)/2a and shows how to use it to solve various quadratic equations by identifying the a, b, and c coefficients and plugging them into the formula. Examples are provided to demonstrate solving equations step-by-step using the quadratic formula.
Department of MathematicsMTL107 Numerical Methods and Com.docx
Department of Mathematics
MTL107: Numerical Methods and Computations
Exercise Set 8: Approximation-Linear Least Squares Polynomial approximation, Chebyshev
Polynomial approximation.
1. Compute the linear least square polynomial for the data:
i xi yi
1 0 1.0000
2 0.25 1.2840
3 0.50 1.6487
4 0.75 2.1170
5 1.00 2.7183
2. Find the least square polynomials of degrees 1,2 and 3 for the data in the following talbe.
Compute the error E in each case. Graph the data and the polynomials.
:
xi 1.0 1.1 1.3 1.5 1.9 2.1
yi 1.84 1.96 2.21 2.45 2.94 3.18
3. Given the data:
xi 4.0 4.2 4.5 4.7 5.1 5.5 5.9 6.3 6.8 7.1
yi 113.18 113.18 130.11 142.05 167.53 195.14 224.87 256.73 299.50 326.72
a. Construct the least squared polynomial of degree 1, and compute the error.
b. Construct the least squared polynomial of degree 2, and compute the error.
c. Construct the least squared polynomial of degree 3, and compute the error.
d. Construct the least squares approximation of the form beax, and compute the error.
e. Construct the least squares approximation of the form bxa, and compute the error.
4. The following table lists the college grade-point averages of 20 mathematics and computer
science majors, together with the scores that these students received on the mathematics
portion of the ACT (Americal College Testing Program) test while in high school. Plot
these data, and find the equation of the least squares line for this data:
:
ACT Grade-point ACT Grade-point
score average score average
28 3.84 29 3.75
25 3.21 28 3.65
28 3.23 27 3.87
27 3.63 29 3.75
28 3.75 21 1.66
33 3.20 28 3.12
28 3.41 28 2.96
29 3.38 26 2.92
23 3.53 30 3.10
27 2.03 24 2.81
5. Find the linear least squares polynomial approximation to f(x) on the indicated interval
if
a. f(x) = x2 + 3x+ 2, [0, 1]; b. f(x) = x3, [0, 2];
c. f(x) = 1
x
, [1, 3]; d. f(x) = ex, [0, 2];
e. f(x) = 1
2
cosx+ 1
3
sin 2x, [0, 1]; f. f(x) = x lnx, [1, 3];
6. Find the least square polynomial approximation of degrees 2 to the functions and intervals
in Exercise 5.
7. Compute the error E for the approximations in Exercise 6.
8. Use the Gram-Schmidt process to construct φ0(x), φ1(x), φ2(x) and φ3(x) for the following
intervals.
a. [0,1] b. [0,2] c. [1,3]
9. Obtain the least square approximation polynomial of degree 3 for the functions in Exercise
5 using the results of Exercise 8.
10. Use the Gram-Schmidt procedure to calculate L1, L2, L3 where {L0(x), L1(x), L2(x), L3(x)}
is an orthogonal set of polynomials on (0,∞) with respect to the weight functions w(x) =
e−x and L0(x) = 1. The polynomials obtained from this procedure are called the La-
guerre polynomials.
11. Use the zeros of T̃3, to construct an interpolating polynomial of degree 2 for the following
functions on the interval [-1,1]:
a. f(x) = ex, b. f(x) = sinx, c. f(x) = ln(x+ 2), d. f(x) = x4.
12. Find a bound for the maximum error of the approximation in Exercise 1 on the interval
[-1,1].
13. Use the zer.
Business Math Chapter 3
The document covers systems of linear equations, including how to solve them using substitution and elimination methods. It provides examples of solving systems of equations with one solution, no solution, and infinitely many solutions. Quadratic equations are also discussed, including how to solve them by factoring, using the quadratic formula, and identifying the nature of solutions based on the discriminant.
Advance algebra
The document provides steps and examples for solving various types of word problems in algebra, including number, mixture, rate/time/distance, work, coin, and geometric problems. It also covers solving quadratic equations using methods like the square root property, completing the square, quadratic formula, factoring, and using the discriminant. Finally, it discusses linear inequalities, including properties related to addition, multiplication, division, and subtraction of inequalities.
Chapter2.3
The document provides examples and explanations for performing arithmetic operations with rational numbers including integers, fractions, and decimals. It demonstrates how to add, subtract, multiply, and divide rational numbers, as well as how to evaluate expressions involving rational numbers. The examples are aligned with California state math standards for performing operations with rational numbers.
Solving systems with elimination
The document provides steps for solving systems of equations using the elimination method with addition and subtraction:
1. Put both equations in standard form.
2. Determine which variable to eliminate by finding terms with the same coefficients.
3. Add or subtract the equations to eliminate the chosen variable.
4. Substitute and solve for the remaining variable and plug back into one equation to solve for the other.
5. Check the solution by substituting the values back into both original equations.
Mathematics 8 Systems of Linear Inequalities
This learner's module discusses and help the students about the topic Systems of Linear Inequalities. It includes definition, examples, applications of Systems of Linear Inequalities.
MODULE 3.pptx
This document discusses arithmetic sequences and finding the nth term of an arithmetic sequence. It provides the formula for finding the nth term: an = a1 + (n-1)d, where an is the nth term, a1 is the first term, d is the common difference, and n is the term position. It works through examples of using the formula to find specified terms like the 21st term or if the 304th term is given. It also provides practice problems for readers to find the nth term of given arithmetic sequences.
Similar to 4.3 cramer’s rule (20)
This quiz is open book and open notes/tutorialoutlet
This quiz is open book and open notes/tutorialoutlet
3 2 solving systems of equations (elimination method)
3 2 solving systems of equations (elimination method)
Department of MathematicsMTL107 Numerical Methods and Com.docx
Department of MathematicsMTL107 Numerical Methods and Com.docx
More from joannahstevens
Linear combination
This document provides instructions on how to solve systems of linear equations using the linear combination method, also known as the elimination method or add/subtract method. It gives examples of using this method to solve systems by eliminating a variable through adding or subtracting the equations. Practice problems are provided for the reader to solve systems of equations using the linear combination method.
Jeopardy review
1. This document contains math word problems and questions about graphing lines, writing equations of lines, finding slopes, and solving inequalities.
2. It asks the learner to summarize slope, graph lines, find slopes of lines, write equations in slope-intercept form, and solve various math problems.
3. The questions cover topics like finding slopes, writing equations, graphing lines and inequalities, finding intercepts, parallel and perpendicular lines, and solving systems of equations.
Introduction to coordinate geometry
Any point on a plane can be named with two numbers called coordinates - an x-coordinate for the distance from the origin along the x-axis and a y-coordinate for the distance along the y-axis. The coordinate grid is divided into four quadrants by the x and y axes, and points can be plotted in the grid and identified by their coordinates and quadrant. Linear equations can be graphed by making a table of solutions and plotting the points, then drawing a straight line through them.
F(x) terminology
This document contains terminology and examples related to functions. It defines f(x) as "the value of y in function f, where the independent variable is x". It provides examples of evaluating functions like f(x) = -7x + 100 for different x values. It also gives examples of composite functions like f(f(x)) and using different functions like f(x) and g(x) together. The examples relate to word problems about distances over time on a boat trip.
Translating words into symbols
This document provides examples of translating word problems into mathematical equations and then solving those equations. It contains sample word problems that translate to equations involving addition, subtraction, multiplication and division. It also includes guidance on common phrases like "less than" and their symbolic representations. Finally, it provides two word problems as examples for students to practice translating words into mathematical symbols.
Inequalities
This document provides information and examples about solving inequalities:
1) It gives examples of solving linear equations with one variable and two variables.
2) It explains the symbols used for different inequality types and how to graph them, including shading the correct side of the inequality.
3) It provides rules for solving inequalities, such as reversing the inequality sign when multiplying or dividing both sides by a negative number.
4) Multiple examples are worked through step-by-step of solving and graphing various inequalities with one variable and multiple variables.
Solving equations with variables on both sides & translating words into symbols
This document contains sample math problems and notes about solving equations with variables on both sides. It includes examples of solving equations like 5y - 3y + 13 = 26, 3s + 2 = 8, 2d + 3d = 15, and 2(w -1) + 8 = 16. The notes provide instructions for solving equations where variables are on both sides, such as 6x = 4x + 18, 3y = 15 - 2y, and 7(a - 2) - 6 = 2a + 8 + a. The document also references a Solving Equations BINGO activity and exit slip homework.
Solving equations
This document provides an introduction to solving equations. It explains the key properties of equality that must be followed when solving equations, such as performing the same operation to both sides of the equal sign. Examples are provided to demonstrate applying these properties to solve one-step, two-step, and multi-step equations. Students are instructed to show their work and given practice problems applying the steps to solve different types of equations.
Transforming equations
This document provides background information on transforming equations and solving one-step equations using addition and subtraction. It explains that if the same number is added to or subtracted from both sides of an equation, the resulting equations will be equal. Examples are provided of solving equations by adding or subtracting the same number from both sides to isolate the variable.
Solving percent equations
The document discusses percentages and solving percentage equations. It provides examples of how to write percentage equations for word problems, such as finding what percentage one number is of another or finding the total given a percentage. It demonstrates solving equations to find percentages and totals in various contexts like test scores, class sizes, sales, and more. Percentages allow for easier comparison of quantities and are used in many everyday situations like grades, weather, nutrition labels, and shopping.
Decimals, fractions, and percents
This document discusses how to convert between decimals, fractions, and percents. It provides examples for converting non-repeating and repeating decimals to fractions by dropping the decimal point and putting the number over the place value. It also demonstrates how to convert fractions to decimals by dividing the numerator by the denominator and moving the decimal point. Additionally, it shows how to convert decimals to percents by moving the decimal point two places to the right and adding the percent sign, and how to convert percents to decimals by moving the decimal point two places to the left and dropping the percent sign.
Notes on 3.2 properties of linear frunction graphs
The document discusses key properties of linear function graphs, including that the y-intercept is the y-value when x=0 and the x-intercept is the x-value when y=0. It defines slope as m in the slope-intercept equation y=mx+b and describes horizontal and vertical lines having slopes of 0 and undefined respectively. It lists three common forms for linear functions: slope-intercept, point-slope, and Ax+By=C and provides examples of transforming between the forms.
Proportions & school pictures
The document discusses proper proportions for various items like wallets, photos, and the American flag. It poses a math problem asking what the width of a 57cm long American flag would need to be in order to have the official proportion of 19 units in length for every 10 units in width.
Inequalities
This document discusses inequalities and provides examples of solving various inequality expressions. It begins by giving an example of a negative number that is not closed under multiplication. It then defines common inequality symbols like less than, greater than, less than or equal to, and greater than or equal to. The document proceeds to give examples of solving linear inequalities assuming the domain is all real numbers, including compound inequalities involving absolute value expressions. It concludes with examples for an exit slip problem set involving solving several inequalities.
The matrix
The document contains information about the reciprocal of 1/9, the value of |-8+-2|, the difference between a January wind chill of -8 degrees and an August heat index of 108 degrees. It also contains a summary of the 1999 film The Matrix and definitions of a matrix and rules for adding, subtracting and multiplying matrices by scalars.
Simplifying expressions, solving, domain and range
This document appears to be notes from a math class covering simplifying expressions, solving equations, and domain and range. It includes examples of irrational and imaginary numbers, naming polynomials by degree and terms, multiplying binomials, homework assignments on the topics, and notes on using the vertical line test to determine if a relation is a function by checking if each x-value has exactly one y-value. Sketching graphs of functions given domains and ranges is also mentioned.
04.07.10 more volume exercises
The document provides instructions to sketch and calculate the volumes of different shapes, including a rectangular prism with dimensions of 3 mm by 5 mm by 15 mm, a cylinder with a diameter of 2.6 m and height of 3.5 m, a rectangular prism with dimensions of 6 inches by 6 inches by 9 inches, and a cylinder with a radius of 3 yards and height of 10 yards. It also gives problems to find the height of a rectangular prism with given length, width, and volume, and to find the radius and height of cylinders with given dimensions and volumes.
04.08.10 applications of volume and surface area
The document contains 4 applications that use surface area and volume concepts:
1) Finding the height of an aquarium tank given its dimensions and volume.
2) Comparing the surface area and volume of a can and box to determine which holds more juice.
3) Calculating the surface area of a gift to see if a sheet of wrapping paper is large enough.
4) Determining which pizza sauce jar is a better buy based on price and volume.
04.07.10 more volume
The document provides instructions for calculating the volumes of different shapes, including a rectangular prism and cylinder. It asks the reader to first predict which solids have the greatest and least volumes without calculations. It then has the reader measure dimensions of different solids to calculate their actual volumes and rank them. Finally, it gives two word problems to find heights using given lengths, widths, radii and volumes.
More from joannahstevens (20)
Solving equations with variables on both sides & translating words into symbols
Solving equations with variables on both sides & translating words into symbols
Notes on 3.2 properties of linear frunction graphs
Notes on 3.2 properties of linear frunction graphs
Simplifying expressions, solving, domain and range
Simplifying expressions, solving, domain and range
4.3 cramer’s rule
- 4. Cramer’s Rule (Section 4.3) Gabriel Cramer was a Swiss mathematician (1704-1752)
- 11. Solution: (-1,2) Example 2