
Examine the consistency of the system of linear equations and
.
Answer
485.1k+ views
Hint: We will be solving this question by writing the given equations into the matrix form
.
For example, we have two equations
and
, so we will write these equations into matrix form
by using the coefficients of each equation to form each row of the matrix as shown below:
.
Complete step-by-step solution:
Step 1: For checking the consistency of the system of linear equations, first of all, we will write the equations into the matrix form
as shown below:
Where,
,
and .
Step 2: Now we will be calculating the determinant of the matrix by using the formula , where , are the elements of the first row and , are the elements of the second one in any matrix.
By using the formula of determinant in , we get:
By solving the brackets in the above expression, we get:
By doing the final subtraction in the RHS side of the above expression, we get:
Which means that . So, if the determinant of the linear equations is not equal to zero then consistency exists.
Therefore, we can say that the system of equations is consistent.
The system of equations is consistent.
Note: Students need to remember the formula for calculating the determinant of the matrix which is denoted by the symbol . So, for any matrix , the determinant of the matrix will be equals to as below:
.
Also, you should remember that if the determinant of the matrix equals zero then the system of equations may be either consistent or inconsistent but if the determinant is non-zero then the system of equations is always consistent.
For example, we have two equations
Complete step-by-step solution:
Step 1: For checking the consistency of the system of linear equations, first of all, we will write the equations into the matrix form
Where,
Step 2: Now we will be calculating the determinant of the matrix
By using the formula of determinant in
By solving the brackets in the above expression, we get:
By doing the final subtraction in the RHS side of the above expression, we get:
Which means that
Therefore, we can say that the system of equations is consistent.
The system of equations is consistent.
Note: Students need to remember the formula for calculating the determinant of the matrix which is denoted by the symbol
Also, you should remember that if the determinant of the matrix equals zero then the system of equations may be either consistent or inconsistent but if the determinant is non-zero then the system of equations is always consistent.
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