Question

Q 12. The number of solutions of the given pair of linear equation 6 x−3 y+10=0and 2 x−y+9=0is

a) no solution b) unique solution c) infinitelymany solution d) none of these​

Answer 1

Qᴜᴇsᴛɪᴏɴ :-

The number of solutions of the given pair of linear equation 6x − 3 y+10 = 0and 2x − y+9 = 0 is

a) no solution b) unique solution c) infinitelymany solution d) none of these

ᴄᴏɴᴄᴇᴘᴛ ᴜsᴇᴅ :-

• A linear equation in two variables represents a straight line in 2D Cartesian plane .

• If we consider two linear equations in two variables, say :-

➻ a1x + b1y + c1 = 0

➻ a2x + b2y + c2 = 0

Then :-

✪ Both the straight lines will coincide if :-

a1/a2 = b1/b2 = c1/c2

➻ In this case , the system will have infinitely many solutions.

➻ If a consistent system has an infinite number of solutions, it is dependent and consistent.

✪ Both the straight lines will be parallel if :-

a1/a2 = b1/b2 ≠ c1/c2.

➻ In this case , the system will have no solution.

➻ If a system has no solution, it is said to be inconsistent.

✪ Both the straight lines will intersect if :-

a1/a2 ≠ b1/b2.

➻ In this case , the system will have an unique solution.

➻ If a system has at least one solution, it is said to be consistent..

____________________

Sᴏʟᴜᴛɪᴏɴ :-

comparing the Given Equations with :-

➻ a1x + b1y + c1 = 0

➻ a2x + b2y + c2 = 0

we get :-

➪ a1 = 6

➪ a2 = 2

➪ b1 = (-3)

➪ b2 = (-1)

➪ c1 = 10

➪ c2 = 9

So, Putting in a1/a2 = b1/b2 = c1/c2 , we get,

➪ (6/2) = (-3)/(-1) = 10/9

➪ (3/1) = (3/1) ≠ (10/9)

➪ a1/a2 = b1/b2 ≠ c1/c2.

Hence, from Above Told concept we can conclude that, Both the straight lines will be parallel and In this case , the system will have no solution. (Option A) .

Answer 2

${\huge{\bf{\red{\underline{Solution:}}}}}$

${\bf{\blue{\underline{Given:}}}}$

$\star \: {\sf{ 6x - 3y + 10 = 0}}$

$\star \: {\sf{ 2x - y + 9= 0}}$

${\bf{\blue{\underline{Now:}}}}$

Compare the. given equations with,

$\star{\sf{ a_{1}x + b_{1} y+ c_{1} = 0 }}$

$\star{\sf{ a_{2} x+ b_{2} y+ c_{2} = 0 }}$

Where,

$: \implies{\sf{ a_{1} = 6}}$

$: \implies{\sf{ a_{2} = 2}}$

$: \implies{\sf{ b_{1} = -3}}$

$: \implies{\sf{ b_{2} = -1}}$

$: \implies{\sf{ c_{1} = 10}}$

$: \implies{\sf{ c_{2} = 9}}$

Here,

$: \implies{\sf{ \frac{6}{2} = \frac{ - 3}{-1} = \frac{10}{9} }}$

$: \implies{\sf{ \frac{3}{1} = \frac{ 3}{ 1} \neq\frac{10}{9} }}$

◕Therefore, the pair of linear equation have no solution.

$\bigstar {\sf\underline{ \orange {\boxed{ Option \: A\: is \: correct}}}}$

_________________________________________

$\bigstar{\sf{\underline{ \green{ \: Unique \: Solution}}}}$

• In this line may intersect in a single point and the pair of equations has a unique solution.

$: \implies{\sf{ \frac{ a_{1} }{ a_{2} } \neq \frac{ b_{1} }{ b_{2} } }}$

$\bigstar{\sf{\underline{ \green{ \: No \: Solution}}}}$

• In this the line may be parallel and the equations have no solution.

$: \implies{\sf{ \frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } \neq\frac{ c_{1} }{ c_{2} } }}$

$\bigstar{\sf{\underline{ \green{ \: Infinitely\:Many \: Solution}}}}$

• In this the line may be conincident and the equation have infinitely many solutions.

$: \implies{\sf{ \frac{ a_{1} }{ a_{2} } = \frac{ b_{1} }{ b_{2} } = \frac{ c_{1} }{ c_{2} } }}$