Example 1
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Gaussian elimination is a method for solving systems of linear equations by transforming the augmented matrix of coefficients into row-echelon form using elementary row operations. The document provides an example of using Gaussian elimination to solve a system of 3 equations by first writing the augmented matrix and then performing a sequence of row operations to obtain the row-echelon form from which the solution can be extracted with back substitution.
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This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using multiplication and addition of equations. Gaussian elimination transforms the coefficient matrix into row echelon form through elementary row operations to solve the system.
System of equations
This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, Gaussian elimination, and Gauss-Jordan elimination. It provides an example of using Cramer's rule to solve a 2x2 system of equations, resulting in solutions of x=2 and y=1. It also gives a step-by-step example of using Gaussian elimination to determine that a given 2x2 system has no solution.
System of equations
This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using multiplication and addition of equations. Gaussian elimination transforms the coefficient matrix into triangular form using row operations, then back substitution can find the unique solution.
System of equations
This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using addition or subtraction of equations. Gaussian elimination transforms the coefficient matrix into row echelon form through elementary row operations to solve the system. The document provides examples of applying each method to solve sample systems of linear equations.
System of equations
This document discusses different methods for solving systems of linear equations, including Cramer's rule, elimination methods, and Gaussian elimination. Cramer's rule uses determinants to find the values of variables by dividing the determinant of the coefficients by the primary determinant. Elimination methods remove one unknown using row operations like addition and subtraction. Gaussian elimination transforms the coefficient matrix into triangular form using row operations, then back substitution can find the unique solution.
chapter7_Sec1.ppt
This document provides an overview of matrices and determinants from a college algebra textbook. It begins with a chapter overview stating that a matrix is a rectangular array of numbers used to organize information. It then discusses representing a linear system of equations as an augmented matrix, which contains the same information as the system in a simpler form. Elementary row operations that can be performed on the augmented matrix are introduced. Gaussian elimination, a method for solving linear systems using the row-echelon form of the augmented matrix and back-substitution, is also covered at a high level.
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic Equations
This powerpoint presentation discusses or talks about the topic or lesson System of Equations involving Quadratic Equations. It also discusses and explains the rules, steps and examples of System of Equations involving Quadratic Equations
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Gauss elimination and Gauss-Jordan elimination are methods for solving systems of linear equations. Gauss elimination puts the matrix of coefficients into row-echelon form using elementary row operations, while Gauss-Jordan elimination further reduces the matrix to reduced row-echelon form. These methods can also be used to find the rank of a matrix, calculate the determinant, and compute the inverse of an invertible square matrix. Examples demonstrate applying the methods to solve systems of 3 equations with 3 unknowns.
Pair of linear equation in two variables (sparsh singh)
The document discusses different methods for solving systems of linear equations:
1) The substitution method involves substituting one variable's expression into the other equation.
2) The elimination method multiples equations to make coefficients equal and subtracts to eliminate one variable.
3) The cross-multiplication method multiplies equations by constants to isolate coefficients and solve for variables.
Each method follows defined steps to systematically solve the system.
Pair of linear equation in two variables (sparsh singh)
The document discusses different methods for solving systems of linear equations:
1) The substitution method involves substituting one variable's expression into the other equation.
2) The elimination method multiples equations to make coefficients equal and subtracts to eliminate one variable.
3) The cross-multiplication method multiplies equations by constants to eliminate one variable, yielding expressions for each variable.
Section-7.4-PC.ppt
This document discusses matrices and their use in solving systems of linear equations. It defines what a matrix is, including its dimensions and entries. It also explains elementary row operations that can be performed on matrices to put them in row echelon form. Gaussian elimination is introduced as a method of using matrices and elementary row operations to solve systems of linear equations through back-substitution. Examples are provided to demonstrate these concepts.
1 Part 2 Systems of Equations Which Do Not Have A Uni.docx
1
Part 2: Systems of Equations Which Do Not Have A Unique
Solution
On the previous pages we learned how to solve systems of equations using Gaussian
elimination. In each of the examples and exercises of part 1(except for exercise 1 parts d and e)
the systems of equations had a unique solution. That is, a single value for each of the variables.
In example 3 we found the solution to be 7 23 3, . This means that the graphs of the two lines in
example 3 intersect at this unique point. In 2-space, the xy-plane, we have the geometric bonus
of being able to draw a picture of the solutions to a system of two equations two unknowns.
Clearly, if we were asked to draw the graphs of two lines in the xy-plane we have 3 basic
choices/cases:
1. Draw the two lines so they intersect. This point of intersection can only happen once for
a given pair of lines. That is, the two lines intersect in a unique point. There is a unique
common solution to the system of equations. Discussed in part 1.
2. Draw the two lines so that one is on "top of" the other. In this case there are an infinite
number of common points, an infinite number of solutions to the given system. Discussed
in part 2.
3. Draw two parallel lines. In this case there are no points common to both lines. There is
no solution to the system of equations that describe the lines. Discussed in part 2.
The 3 cases above apply to any system of equations.
Theorem 1. For any system of m equations with n unknowns (m < n) one of the following cases
applies:
1. There is a unique solution to the system.
2. There is an infinite number of solutions to the system.
3. There are no solutions to the system.
Again, in this section of the notes we will illustrate cases 2 and 3. To solve systems of
equations where these cases apply we use the matrix procedure developed previously.
Example 6. Solve the system
x + 2y = 1
2x + 4y = 2
2
It is probably already clear to the reader that the second equation is really the first in
disguise. (Simply divide both sides of the second equation by 2 to obtain the first). So if we
were to draw the graph of both we would obtain the same line, hence have an infinite number of
points common to both lines, an infinite number of solutions. However it would be helpful in
solving other systems where the solutions may not be so apparent to do the problem
algebraically, using matrices. The matrix of the system with its simplification follows. Recall,
we try to express the matrix
1 2 1
2 4 2
in the form 1
2
1 0
0 1
b
b
from which we can read off the
solution. However after one step we note that
1 2 1
2 4 2
1 22 R R
1 2 1
0 0 0
. It should be clear to the reader that no matter what further
elementary row operations we perform on the matrix
1 2 1
0 0 0
we cannot change it to the form
we hoped for, namel.
.Chapter7&8.
This document discusses systems of linear and quadratic equations. It begins by introducing systems of two linear equations in two variables and methods for solving them, like substitution and elimination. It then covers systems of linear equations in three variables, whose solution sets are intersections of planes. An example demonstrates the substitution method for a 3-variable system. Later sections discuss systems involving quadratic equations, providing an example that solves a linear-quadratic system by substituting one variable in terms of the other. Exercises provide additional examples of solving various systems of equations.
9.3 Solving Systems With Gaussian Elimination
Write the augmented matrix of a system of equations.
Write the system of equations from an augmented matrix.
Perform row operations on a matrix.
Solve a system of linear equations using matrices.
Systems of Linear Equations
The document discusses methods for solving systems of linear equations in two variables, including graphing the lines defined by each equation to find their point of intersection, the elimination method of adding multiples of equations to remove a variable, and the substitution method of solving one equation for one variable and substituting it into the other equation. It provides examples of applying these different solution methods to solve systems of linear equations and determining if a solution exists.
7.2 Systems of Linear Equations - Three Variables
This document discusses solving systems of three linear equations with three variables. It introduces Gaussian elimination as a method for solving such systems. Gaussian elimination involves eliminating variables one at a time to solve for the remaining variables in reverse order. The document provides an example of using Gaussian elimination to solve a 3x3 system. It also discusses the different outcomes possible for 3x3 systems, such as a single solution, infinitely many solutions, or no solution if the system is inconsistent or dependent.
Lca10 0505
This document discusses solving nonlinear systems of equations. It provides 6 examples of solving nonlinear systems using various methods like substitution, elimination, and a combination of methods. It also discusses how to visualize the graphs of nonlinear systems and how complex solutions may arise. The final example uses a nonlinear system to find the dimensions of a box given the volume and surface area. Key methods taught are substitution, elimination, factoring, and using the quadratic formula.
Similar to Example 1 (20)
GATE Engineering Maths : System of Linear Equations
GATE Engineering Maths : System of Linear Equations
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic Equations
Mathematics 9 Lesson 1-D: System of Equations Involving Quadratic Equations
Pair of linear equation in two variables (sparsh singh)
Pair of linear equation in two variables (sparsh singh)
Pair of linear equation in two variables (sparsh singh)
Pair of linear equation in two variables (sparsh singh)
1 Part 2 Systems of Equations Which Do Not Have A Uni.docx
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Example 1
- 1. Gaussian Elimination Image from WikiMedia Commons created in LightWave by stib http://commons.wikimedia.org/wiki/File:Secretsharing-3-point.png
- 5. Example