Mathematics is a kind of practical subject which is never going to be outdated, but when students come in twelfth grade, the math problems become get more complex. To evaluate the way of solving these problems students have to implement all the math principles which they have learned in their previous grades. For students, grade XII is the stage where their actual math skills are being tested. For more on visit this
Today we are going to talk about an important topic of 12th grade algebra which is “ Types of Linear System”. Let's start with the topic of Algebra problems that are included in 12th grade math, they are basically divided in three parts, Algebra 1, Algebra 2 and linear algebra. Our topic today is a part of linear algebra and it involves a math problem in which the student has to manipulate more than one equation to solve the system. When all the equations of the system are in linear form than it may be a little easier to solve but the real challenge is solving the ones which are of non-linear form! Let us take an example of linear system:
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Now all the equations above are of linear form with three variables. While solving linear system one point should always be kept in mind is that all the equations must have all variables in it as in above equations each of them includes x, y and z.
Suppose if any of the variable is missing in any equation then we have to assume it on our own as
Ax + z= c here y's derivative was missing and the equation can be rewritten as: Ax + 0(y) + z = c
This is done to make easier the comparison of all the equations in the system.
The solution of linear system is done by implementing two standard applications which are Gaussian Elimination and Gauss-Jordan method. Both of these methods prefer the matrix conversion of the linear system and ones the equations are converted into matrix form than the solution is found by calculating the rank of the matrix, determinant of the matrix and by inverse of the matrix.
Let us see how these fundamentals are executed on a linear system:
Firstly convert the linear system into matrix form, the formed matrix includes the integer coefficients of all the variables used to form that system:
1 1 1 | 0
1 -2 2 | 4
1 2 -1| 2
In above matrix the first column represents the value of x coefficient of all three equations, similarly second column represents y's value and third column is for z's value. The fourth column which is separated from a line is the value of the equations in their right side. To solve this linear system the Gaussian Algorithm defines some steps which are as:
to convert matrix into triangular form:
first: eliminate x from every row accept first one.
Second: eliminate y from every row below second row.
So now apply this steps on the above matrix as:
To eliminate x from 2nd and 3rd row you have to perform 2 elimination steps which are:
r3 → r3 – r1
here 'r' represents the row,
now r2-> r2 – r1
The above matrix form shows the proper executed form of first step of algorithm because all x values are removed from 2nd and 3rd rows.
Now for executing second step of the algorithm we have to perform following :
r2 → (r2 x 3) + r3
The above matrix form represents a triangular matrix, which is the resulted matrix of Gaussian algorithm. Now this matrix is further converted into linear equation form whose solution is much easy in comparison of the given linear system.
So converted linear system from Gaussian elimination is as:
The determination of all unknown variables x, y and z is now being done by solving these linear equations as:
x + y + z = 0....... 1
-3y + z = 4...........2
-5z = 10.........3
from the third equation
z= -10/ 5 = -2
Now put this value of z in equation number 2:
-3y + (-2) = 4
-3y = 6
-y = 6/-3 = -2
Use the value of 'y and 'z' to evaluate the value of 'x'.
x + (-2) + (-2)= 0
x = 4
So the solution matrix of the linear system is as;
x = 4
y = -2
z = -2
In Gaussian Elimination the solution of the linear system depends on the matrix Row elimination system but the other application Gauss-Jordan, prefers the matrix inverse program to normalize the system of equations into more simpler form by which the value of unknown variables in the equation easily can be calculated.
Matrix is an essential tool for 12th grade math problems because various complex problems are solved by implementing their related fundamentals and principles on corresponding matrix form. That's why to solve linear system and similar problems involving long calculations, students have to learn various matrix operations. Here we are going to elaborate some of the important matrix operations which are RANK determination and inverse matrix. First let us see what RANK determination actually means for a matrix?, Every student is aware of a matrix formation which includes certain number of rows and columns in it. If there are 'm' rows and 'n' columns in any matrix then the rank of the matrix is decided based on following rules. If both m and n are equal in other words, m = n than RANK is the equal to m or n. But if m>n than rank is n and vice versa. It means that the lowest value among the rows and column represents the RANK of the matrix. By calculating rank, number of solutions gets clarified for any linear system.
Now second most commonly applied operation on matrix is Inverse matrix calculations, if A is a matrix than A-1 represents its inverse form. To convert any matrix into its inverse form the use of Identity matrix is preferred.
Identity matrix is represented as [1 0]
[0 1]
If A is of m x n than an identity matrix of same dimensions is merged with it as:
AX= 1 3 3 | 1 0 0
1 4 3 | 0 1 0
1 3 4 | 0 0 1
the target is to convert the left side of the dashed line into identity by applying various row transformation as:
1 3 3 | 1 0 0 - R1 + R2 1 3 3 | 1 0 0
1 4 3 | 0 1 0 --------------> 0 1 0 | -1 1 0
1 3 4 | 0 0 1 - R1 + R2 0 0 1 | -1 0 1
-3 R2 + R1 ---------------------> 1 0 3 | 4 -3 0
0 1 0 | -1 1 0
0 0 1 | -1 0 1
-3 R3+ R1 ----------------------> 1 0 0 | 7 -3 -3
0 1 0 | -1 1 0
0 0 1 | -1 0 1
The right side of the final matrix represents the Inverse of the A matrix. To learn more matrix operations which are required to solve various types of linear system and other linear algebra problems students can rely on various online math tutoring services. These services deal with every student of various standards. The required term to be good in math is practice. If you are doing regular math practice than that will definitely excel your math skills and for proceeding help in this task, online tutors avail various math queries work sheets for students and also give their assistance in solving them whenever required. These online tutors make students to learn the optimized way of solving any problem which helps them to solve any complex query within a short while. In present time to move into college or in other higher education system , students have to go through various competitive entrance exams whose syllabus mostly includes 11th and 12th grade math queries. So if students are not able to sort out the queries in a short time than these exams become the toughest task to go through with. Because in these kind of exams the number of questions to be solved are large and time given to solve them is not enough if you preferred the standard implementation. The 12th grade includes vast math syllabus in which all math branches are scheduled into various units. The major part of these problems is that you need to convert every problem in such form where from you can apply your math standard principles on them to sort out. We have discussed above how to evaluate the problems of linear systems but this clause does not end here. The question arises “ when the equations of the system are not in linear form then how will be they solved?”. In that situation students need to implement various suitable Normalization techniques to convert non-linear equations into their corresponding linear form. To learn more about non-linear types of linear system students can attend the regular math sessions organized by Online math tutoring services.
In upcoming posts we will discuss about Syllabus of XII Grade and Discrete and continuous random variables. Visit our website for information on West Bengal class 12 syllabus
Today we are going to talk about an important topic of 12th grade algebra which is “ Types of Linear System”. Let's start with the topic of Algebra problems that are included in 12th grade math, they are basically divided in three parts, Algebra 1, Algebra 2 and linear algebra. Our topic today is a part of linear algebra and it involves a math problem in which the student has to manipulate more than one equation to solve the system. When all the equations of the system are in linear form than it may be a little easier to solve but the real challenge is solving the ones which are of non-linear form! Let us take an example of linear system:
Now all the equations above are of linear form with three variables. While solving linear system one point should always be kept in mind is that all the equations must have all variables in it as in above equations each of them includes x, y and z.
Suppose if any of the variable is missing in any equation then we have to assume it on our own as
Ax + z= c here y's derivative was missing and the equation can be rewritten as: Ax + 0(y) + z = c
This is done to make easier the comparison of all the equations in the system.
The solution of linear system is done by implementing two standard applications which are Gaussian Elimination and Gauss-Jordan method. Both of these methods prefer the matrix conversion of the linear system and ones the equations are converted into matrix form than the solution is found by calculating the rank of the matrix, determinant of the matrix and by inverse of the matrix.
Let us see how these fundamentals are executed on a linear system:
Firstly convert the linear system into matrix form, the formed matrix includes the integer coefficients of all the variables used to form that system:
1 1 1 | 0
1 -2 2 | 4
1 2 -1| 2
In above matrix the first column represents the value of x coefficient of all three equations, similarly second column represents y's value and third column is for z's value. The fourth column which is separated from a line is the value of the equations in their right side. To solve this linear system the Gaussian Algorithm defines some steps which are as:
to convert matrix into triangular form:
first: eliminate x from every row accept first one.
Second: eliminate y from every row below second row.
So now apply this steps on the above matrix as:
To eliminate x from 2nd and 3rd row you have to perform 2 elimination steps which are:
r3 → r3 – r1
here 'r' represents the row,
now r2-> r2 – r1
The above matrix form shows the proper executed form of first step of algorithm because all x values are removed from 2nd and 3rd rows.
Now for executing second step of the algorithm we have to perform following :
r2 → (r2 x 3) + r3
The above matrix form represents a triangular matrix, which is the resulted matrix of Gaussian algorithm. Now this matrix is further converted into linear equation form whose solution is much easy in comparison of the given linear system.
So converted linear system from Gaussian elimination is as:
The determination of all unknown variables x, y and z is now being done by solving these linear equations as:
x + y + z = 0....... 1
-3y + z = 4...........2
-5z = 10.........3
from the third equation
z= -10/ 5 = -2
Now put this value of z in equation number 2:
-3y + (-2) = 4
-3y = 6
-y = 6/-3 = -2
Use the value of 'y and 'z' to evaluate the value of 'x'.
x + (-2) + (-2)= 0
x = 4
So the solution matrix of the linear system is as;
x = 4
y = -2
z = -2
In Gaussian Elimination the solution of the linear system depends on the matrix Row elimination system but the other application Gauss-Jordan, prefers the matrix inverse program to normalize the system of equations into more simpler form by which the value of unknown variables in the equation easily can be calculated.
Matrix is an essential tool for 12th grade math problems because various complex problems are solved by implementing their related fundamentals and principles on corresponding matrix form. That's why to solve linear system and similar problems involving long calculations, students have to learn various matrix operations. Here we are going to elaborate some of the important matrix operations which are RANK determination and inverse matrix. First let us see what RANK determination actually means for a matrix?, Every student is aware of a matrix formation which includes certain number of rows and columns in it. If there are 'm' rows and 'n' columns in any matrix then the rank of the matrix is decided based on following rules. If both m and n are equal in other words, m = n than RANK is the equal to m or n. But if m>n than rank is n and vice versa. It means that the lowest value among the rows and column represents the RANK of the matrix. By calculating rank, number of solutions gets clarified for any linear system.
Now second most commonly applied operation on matrix is Inverse matrix calculations, if A is a matrix than A-1 represents its inverse form. To convert any matrix into its inverse form the use of Identity matrix is preferred.
Identity matrix is represented as [1 0]
[0 1]
If A is of m x n than an identity matrix of same dimensions is merged with it as:
AX= 1 3 3 | 1 0 0
1 4 3 | 0 1 0
1 3 4 | 0 0 1
the target is to convert the left side of the dashed line into identity by applying various row transformation as:
1 3 3 | 1 0 0 - R1 + R2 1 3 3 | 1 0 0
1 4 3 | 0 1 0 --------------> 0 1 0 | -1 1 0
1 3 4 | 0 0 1 - R1 + R2 0 0 1 | -1 0 1
-3 R2 + R1 ---------------------> 1 0 3 | 4 -3 0
0 1 0 | -1 1 0
0 0 1 | -1 0 1
-3 R3+ R1 ----------------------> 1 0 0 | 7 -3 -3
0 1 0 | -1 1 0
0 0 1 | -1 0 1
The right side of the final matrix represents the Inverse of the A matrix. To learn more matrix operations which are required to solve various types of linear system and other linear algebra problems students can rely on various online math tutoring services. These services deal with every student of various standards. The required term to be good in math is practice. If you are doing regular math practice than that will definitely excel your math skills and for proceeding help in this task, online tutors avail various math queries work sheets for students and also give their assistance in solving them whenever required. These online tutors make students to learn the optimized way of solving any problem which helps them to solve any complex query within a short while. In present time to move into college or in other higher education system , students have to go through various competitive entrance exams whose syllabus mostly includes 11th and 12th grade math queries. So if students are not able to sort out the queries in a short time than these exams become the toughest task to go through with. Because in these kind of exams the number of questions to be solved are large and time given to solve them is not enough if you preferred the standard implementation. The 12th grade includes vast math syllabus in which all math branches are scheduled into various units. The major part of these problems is that you need to convert every problem in such form where from you can apply your math standard principles on them to sort out. We have discussed above how to evaluate the problems of linear systems but this clause does not end here. The question arises “ when the equations of the system are not in linear form then how will be they solved?”. In that situation students need to implement various suitable Normalization techniques to convert non-linear equations into their corresponding linear form. To learn more about non-linear types of linear system students can attend the regular math sessions organized by Online math tutoring services.
In upcoming posts we will discuss about Syllabus of XII Grade and Discrete and continuous random variables. Visit our website for information on West Bengal class 12 syllabus
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