Question

3x-4y=7 9x+12y=14 determine whether the system of linear equation has unique solution, infinitely many solution or no solution

Answer 1

Answer:

• Given system of equations has unique solution.

Explanation:

Knowledge required:

For a system of linear equations in two variables

• $\sf{a_1x+b_1y+c_1=0}$  and

• $\sf{a_2x+b_2y+c_2=0}$

Equations will have a unique solution and represent intersecting lines when,

• $\sf{\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}}$

Equations will have infinitely many solutions and represent coincident lines when,

• $\sf{\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}}$

Equations will have no solution and represent parallel lines when,

• $\sf{\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}}$

Solution:

Given equations are:  

$\sf{3x-4y=7\;\;and\;\;9x+12y=14}$

They can be written as:

$\sf{3x-4y-7=0\;\;and\;\;9x+12y-14=0}$

On comparing with $\sf{a_1x+b_1y+c_1=0\;\;and\;\;a_2x+b_2y+c_2=0}$ we will get,

• $\sf{a_1=3\;\;;b_1=-4\;\;;c_1=-7}$

• $\sf{a_2=9\;\;;b_2=12\;\;;c_2=-14}$

Now comparing ratio

$\implies\sf{\dfrac{a_1}{a_2}=\dfrac{3}{9}=\dfrac{1}{3}}$

$\implies\sf{\dfrac{b_1}{b_2}=\dfrac{-4}{12}=\dfrac{-1}{3}}$

$\implies\sf{\dfrac{c_1}{c_2}=\dfrac{-7}{-14}=\dfrac{1}{2}}$

So, we got

$\boxed{\sf{\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}}}$

Therefore,

• System of linear equations has unique solution.