Question

1. If the pair of Linear equations x + 2y = 3 and 2x + 4y = k are coincide then the
value of 'k' is :
A. 3
B. 6
C.
-3
D. -6

Answer 1

ANSWER:

Given:

• x + 2y = 3 and 2x + 4y = k

To Find:

• Value of such that the linear equations coincide

Concept:

$\text{Let there be a pair of linear equations,}\\\\:\hookrightarrow a_1x+b_1y=c_1\:\:\:\&\:\:\:a_2x+b_2y=c_2\\\\\text{We know that, for a pair of linear equations}\\\\\text{The ratio of corresponding coefficients of variables and constant terms gives following results:}\\\\1):\longrightarrow If\:\:\dfrac{a_1}{a_2}\neq\dfrac{b_1}{b_2}\\\\:\implies\text{Then, 1 solution exist and graphically they intersect with each other.}$

$2):\longrightarrow If\:\:\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\\\\:\implies\text{Then, infinite solutions exist and graphically they coincide with each other.}\\\\3):\longrightarrow If\:\:\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}\neq\dfrac{c_1}{c_2}\\\\:\implies\text{Then, 0 solutions exist and graphically they are parallel to each other.}$

Solution:

$\text{We are given that, the pair of linear equations coincide with each other.}\\\\\text{So, we will take the 2nd result.}\\\\:\hookrightarrow\:\:\dfrac{a_1}{a_2}=\dfrac{b_1}{b_2}=\dfrac{c_1}{c_2}\\\\\text{Here,}\\\\:\longrightarrow a_1=1,\,a_2=2,\,b_1=2,\,b_2=4,\,c_1=3,\,c_2=k\\\\\text{So,}\\\\:\implies\:\: \dfrac{1}{2}=\dfrac{2}{4}=\dfrac{3}{k}\\\\:\implies\:\:\dfrac{1}{2}=\dfrac{3}{k}\\\\:\implies\:\:k=3\times2\\\\\bf{:\implies k=6}\\\\\text{\bf{Hence, for k = 6, the equations coincide with each other.}}$