Question

For what value of k’ the system has infinitely many solution

2y = 4x - 6 & kx = y + 3​

Answer 1

Solution:-

Given equation is

$\rm \: 4x - 2y - 6 = 0 \: \: \: \: ...(i)eq$

$\rm \: kx - y - 3 = 0 \: \: \: \: ......(ii)eq$

These equation are of the form

$\rm \: a_1x + b_1y + _1 = 0 \: \: \: \: \: \: and$

$\rm \: a_2x + b_2y + c_2 = 0$

where

$\rm \: a_1 = 4 ,b_1 = - 2 \: and \: c_1 = - 6$

$\rm \: a_2 = k ,b_2 = - 1 \: and \: c_2 = - 3$

Let the given system of equation have infinitely many solutions

$\rm \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Then

$\rm \: \frac{4}{k} = \frac{ - 2}{ - 1} = \frac{ - 6}{ - 3}$

Taking

$\rm \: \frac{4}{k} = \frac{2}{1}$

using cross multiplication

$\rm \: 2k = 4$

$\rm \: k = \frac{4}{2}$

$\rm \: k = 2$

so the value of k =2

More information

Consistent and inconsistent system of linear equations

=> A system of the equation

$\rm \: a_1x + b_1y + _1 = 0 \: \: \: \: \: \: and \: \: \: \rm \: a_2x + b_2y + c_2 = 0$

is said to be consistent if it has at least one solution. on the other hand , the above system is said to be inconsistent if ot has no solution at all