Show that the following systems of linear equations is consistent and also find their solution:
Answer
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Hint: Consistent linear equations are those equations if it contains a unique solution or infinitely many solutions then it will come under the consistent equation. In this question, we will find the determinant which will be and if all of them are the same then it will come under the consistent equation.
Complete step-by-step answer:
So we have the equation given as
Now we will check the value of and then we will come to know whether the system of linear equations is consistent or not.
So forming the determinant for , it will be as
Now on solving the above determinant we will get
Now by doing the cross multiplication, we have
And on solving it we get
Since the determinant is zero then it means that two of its rows or columns are the same. So by using this property we can directly write that the determinant will be zero.
Now, forming the determinant for , it will be as
For this just changing any one row or column with the constant, then the determinant will be
And it can also be written as
Again we can see that two of its rows or columns are the same. So by using this property we can directly write that the determinant will be zero.
Similarly for
And it can also be written as
Therefore,
Now for , it will be
Therefore,
Since so the given system will be consistent as it has an infinite number of solutions.
Now for finding the solutions, Let us assume will be equal to
Then,
And the other equation formed will be
Now on solving both of this equation, but for this, we have to keep the constant to one side so we get
and we will name it equation
and we will name it equation
Now on solving both of the equation by making one term similar, we will get the values as
and
Hence, we can say that the system of linear equations is consistent having the solution as and .
So, the correct answer is “ , and .”.
Note: This type of question is very simple if we know how to calculate the determinant we can easily solve such questions. System of linear equations can be used in our daily life. Like economists use it to compare gain and loos by graphs, similarly, engineers use it for comparing the mortgages.
Complete step-by-step answer:
So we have the equation given as
Now we will check the value of
So forming the determinant for
Now on solving the above determinant we will get
Now by doing the cross multiplication, we have
And on solving it we get
Since the determinant is zero then it means that two of its rows or columns are the same. So by using this property we can directly write that the determinant will be zero.
Now, forming the determinant for
For this just changing any one row or column with the constant, then the determinant will be
And it can also be written as
Again we can see that two of its rows or columns are the same. So by using this property we can directly write that the determinant will be zero.
Similarly for
And it can also be written as
Therefore,
Now for
Therefore,
Since
Now for finding the solutions, Let us assume
Then,
And the other equation formed will be
Now on solving both of this equation, but for this, we have to keep the constant to one side so we get
Now on solving both of the equation by making one term similar, we will get the values as
Hence, we can say that the system of linear equations is consistent having the solution as
So, the correct answer is “
Note: This type of question is very simple if we know how to calculate the determinant we can easily solve such questions. System of linear equations can be used in our daily life. Like economists use it to compare gain and loos by graphs, similarly, engineers use it for comparing the mortgages.
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