What is the condition for a unique solution of a pair of linear equations in two variables?
Answer
361.5k+ views
Hint: In this question first assume any pair of linear equations in two variables and convert them into matrix format so use these concepts to reach the solution of the question.
Complete step-by-step answer:
Let us consider the system of linear equations in two variables be
$
ax + by = c \\
dx + ey = f \\
$
Where (x, y) are the variables and (a, b, c, d, e, f) are the constants.
Now convert this system of equation into matrix format we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
a&b \\
d&e
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
x \\
y
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
c \\
f
\end{array}} \right]$
In above matrix format determinant (D) = $\left| {\begin{array}{*{20}{c}}
a&b \\
d&e
\end{array}} \right|$
So, the condition of unique solution of a pair of linear equation in two variables is
The value of determinant (D) should not equal zero.
$ \Rightarrow D \ne 0$
Or,
$D = \left| {\begin{array}{*{20}{c}}
a&b \\
d&e
\end{array}} \right| \ne 0$
Expand the determinant we have
$D = \left( {ae - bd} \right) \ne 0$
So, this is the required condition of a unique solution of a pair of linear equations in two variables.
So, this is the required answer.
Note: In such types of questions first let any two linear equations as above then convert it into matrix format as above and calculate the value of determinate so, for unique solution the value of determinant should not equal to zero if zero then the system of equations has either no solution or infinitely many solutions.
Complete step-by-step answer:
Let us consider the system of linear equations in two variables be
$
ax + by = c \\
dx + ey = f \\
$
Where (x, y) are the variables and (a, b, c, d, e, f) are the constants.
Now convert this system of equation into matrix format we have,
$ \Rightarrow \left[ {\begin{array}{*{20}{c}}
a&b \\
d&e
\end{array}} \right]\left[ {\begin{array}{*{20}{c}}
x \\
y
\end{array}} \right] = \left[ {\begin{array}{*{20}{c}}
c \\
f
\end{array}} \right]$
In above matrix format determinant (D) = $\left| {\begin{array}{*{20}{c}}
a&b \\
d&e
\end{array}} \right|$
So, the condition of unique solution of a pair of linear equation in two variables is
The value of determinant (D) should not equal zero.
$ \Rightarrow D \ne 0$
Or,
$D = \left| {\begin{array}{*{20}{c}}
a&b \\
d&e
\end{array}} \right| \ne 0$
Expand the determinant we have
$D = \left( {ae - bd} \right) \ne 0$
So, this is the required condition of a unique solution of a pair of linear equations in two variables.
So, this is the required answer.
Note: In such types of questions first let any two linear equations as above then convert it into matrix format as above and calculate the value of determinate so, for unique solution the value of determinant should not equal to zero if zero then the system of equations has either no solution or infinitely many solutions.
Last updated date: 29th Sep 2023
•
Total views: 361.5k
•
Views today: 8.61k
Recently Updated Pages
What do you mean by public facilities

Difference between hardware and software

Disadvantages of Advertising

10 Advantages and Disadvantages of Plastic

What do you mean by Endemic Species

What is the Botanical Name of Dog , Cat , Turmeric , Mushroom , Palm

Trending doubts
How do you solve x2 11x + 28 0 using the quadratic class 10 maths CBSE

Summary of the poem Where the Mind is Without Fear class 8 english CBSE

The poet says Beauty is heard in Can you hear beauty class 6 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Difference Between Plant Cell and Animal Cell

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

What is the past tense of read class 10 english CBSE

The equation xxx + 2 is satisfied when x is equal to class 10 maths CBSE

Differentiate between homogeneous and heterogeneous class 12 chemistry CBSE
