Question

The value(s) of k for which the pair of linear equations 3x - 2y – 7 = 0 and 6x + ky + 11 = 0 have a unique solution is/are _________________. *

4

All real numbers except 4

-4

All real numbers except -4

​

Answer 1

$\huge\boxed{\underline{\underline{\green{Solution}}}} </p><p>$

$\displaystyle \: \longmapsto \: \:$FORMULA TO BE IMPLEMENTED :

A pair of Straight Lines

$\displaystyle \: a_1x+b_1y+c_1=0 \: and \: \: a_2x+b_2y+c_2=0$

have Unique Solution if $\displaystyle \: \: \frac{a_1}{a_2} \neq \frac{b_1}{b_2}$

$\displaystyle \: \longmapsto \: \:$CALCULATION :

Given pair of linear equations

$3x - 2y – 7 = 0 \: \: and \: \: 6x + ky + 11 = 0$

Comparing with

$\displaystyle \: a_1x+b_1y+c_1=0 \: and \: \: a_2x+b_2y+c_2=0$

We get

$\displaystyle \: a_1 = 3 \: , \: b_1 = - 2 \: , c_1= - 7 \: and \: \: a_2 = 6 \: , \: b_2 = k \: , \: \: c_2=11$

So the given pair of Straight Lines have Unique Solution if

$\displaystyle \: \: \frac{3}{6} \neq \frac{ - 2}{k}$

$\implies \: \: \displaystyle \: 3k \: \ne \: - 12$

$\implies \: \: \displaystyle \: k \: \ne \: - 4$

$\displaystyle \: \longmapsto \: \:$RESULT :

Hence the given Pair of Straight Lines have Unique Solution for all Real Numbers Except - 4

Answer 2

Answer:

A pair of Straight Lines

\displaystyle \: a_1x+b_1y+c_1=0 \: and \: \: a_2x+b_2y+c_2=0a1x+b1y+c1=0anda2x+b2y+c2=0

have Unique Solution if \displaystyle \: \: \frac{a_1}{a_2} \neq \frac{b_1}{b_2}a2a1=b2b1

\displaystyle \: \longmapsto \: \:⟼ CALCULATION :

Given pair of linear equations

3x - 2y – 7 = 0 \: \: and \: \: 6x + ky + 11 = 03x−2y–7=0and6x+ky+11=0

Comparing with

\displaystyle \: a_1x+b_1y+c_1=0 \: and \: \: a_2x+b_2y+c_2=0a1x+b1y+c1=0anda2x+b2y+c2=0

We get

\displaystyle \: a_1 = 3 \: , \: b_1 = - 2 \: , c_1= - 7 \: and \: \: a_2 = 6 \: , \: b_2 = k \: , \: \: c_2=11a1=3,b1=−2,c1=−7anda2=6,b2=k,c2=11

So the given pair of Straight Lines haveUnique Solution if

\displaystyle \: \: \frac{3}{6} \neq \frac{ - 2}{k}63=k−2

\implies \: \: \displaystyle \: 3k \: \ne \: - 12⟹3k=−12

\implies \: \: \displaystyle \: k \: \ne \: - 4⟹k=−4

\displaystyle \: \longmapsto \: \:⟼ RESULT :

Hence the given Pair of Straight Lineshave Unique Solution for all RealNumbers Except - 4