Aptitude Overflow
0 votes
80 views

For what values of $‘k’$ will the pair of equations $3x + 4y =12$ and $kx + 12y = 30$ $NOT$ have a unique solution ?

  1. $9$
  2. $12$
  3. $3$
  4. $7.5$
in Quantitative Aptitude by (4.6k points) 4 31 121
edited by | 80 views

2 Answers

+1 vote

Option A. 9

To not have a unique solution, lines should be parallel.

Two parallel lines have equal slopes.

Slope of the line 3x+4y = 12, = -3/4

Slope of the line kx+12y = 30, = -k/12
3/4 = k/12 or k=9

by (366 points) 3 15
0 votes

A system of linear equations $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$ will have a unique solution if the two lines represented by the equations $a_{1}x + b_{1}y + c_{1} = 0$ and $a_{2}x + b_{2}y + c_{2} = 0$ intersect at a point.
i.e., if the two lines are neither parallel nor coincident. Essentially, the slopes of the two lines should be different.

  • Condition for unique solution$:\dfrac{a_{1}}{a_{2}} \neq \dfrac{b_{1}}{b_{2}}$
  • Condition for no solution$:\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} \neq \dfrac{c_{1}}{c_{2}}$
  • Condition for infinitely many solutions$:\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} = \dfrac{c_{1}}{c_{2}}$

Now, question asked about not unique solution ,the the condition will be $\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}}$

Given the equations, $3x + 4y =12$ and $kx + 12y = 30$

Here, $a_{1} = 3,b_{1} = 4,a_{2} = k,b_{2} = 12$

Now, if above equation does not have unique solution , then $\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}}$

$\implies \dfrac{3}{k} = \dfrac{4}{12}$

$\implies k = 9.$

So, the correct answer is $(A).$

by (4.6k points) 4 31 121
edited by
Answer:

Related questions

Quick search syntax
tags tag:apple
author user:martin
title title:apple
content content:apple
exclude -tag:apple
force match +apple
views views:100
score score:10
answers answers:2
is accepted isaccepted:true
is closed isclosed:true
4,624 questions
1,620 answers
534 comments
44,549 users