# CAT Quantitative Aptitude Questions | CAT Algebra - Linear Equations; Quadratic Equations

###### CAT Questions | Algebra | Linear Equations - Unique Solutions CAT Questions

The question is about checking for consistency of the given equations. Knowing a few equations, we can find whether the equations have an unique solution or infinite solutions or no solution. Framing and solving equations is an integral part of Linear Equations and Quadratic Equations. Get as much practice as you can in these two topics because the benefits of being good at framing equations will be useful in other concepts.

Question 9: a1x + b1y + c1z = d1, a2x + b2y + c2z = d2, a3x + b3y + c3z = d3.

Which of the following statements if true would imply that the above system of equations does not have a unique solution?
i. $$frac{a_{1}}{a_{2}}\\$ = $\frac{b_{1}}{b_{2}}\\$ = $\frac{c_{1}}{c_{2}}\\$ ≠ $\frac{d_{1}}{d_{2}}\\$ ii. $\frac{ a_{1} }{ a_{2} }\\$= $\frac{ a_{2} }{ a_{3} }\\$ ; $\frac{ b_{1} }{ b_{2} }\\$= $\frac{ b_{2} }{ b_{3} }\\$ iii. a1, a2, a3 are integers; b1, b2, b3 are rational numbers, c1, c2, c3 are irrational numbers 1. Statement i 2. Statement ii 3. Statement iii 4. None ## Best CAT Online Coaching Try upto 40 hours for free Learn from the best! #### 2IIM : Best Online CAT Coaching. ### Video Explanation ## Best CAT Coaching in Chennai #### CAT Coaching in Chennai - CAT 2022Limited Seats Available - Register Now! ### Explanatory Answer ##### Method of solving this CAT Question from CAT Algebra - Linear Equations: How we would find out whether the equations are independent? If we have three independent equations, we will have a unique solution. In other words, we will not have unique solutions if: The equations are inconsistent or Two equations can be combined to give the third Now, let us move to the statements. i.$\frac{ a_{1} }{ a_{2} }\\$ = $\frac{ b_{1} }{ a_{2} }\\$ = $\frac{ c_{1} }{ c_{2} }\\$ ≠ $\frac{ d_{1} }{ d_{2} }\\$ This tells us that the first two equations cannot hold good at the same time. Think about this: x + y + z = 3; 2x + 2y + 2z = 5. Either the first or the second can hold good. Both cannot hold good at the same time. So, this will definitely not have any solution. ii.$\frac{ a_{1} }{ a_{2} }\\$= $\frac{ a_{2} }{ a_{3} }\\$ and $\frac{ b_{1} }{ b_{2} }\\$ = $\frac{ b_{2} }{ b_{3} }\\$ a1, a2, a3 are in GP, b1, b2, b3. This does not prevent the system from having a unique solution. For instance, if we have x + 9y + 5z = 11 2x + 3y – 6z = 17 4x + y – 3z = 15 This could very well have a unique solution. iii. a1, a2, a3 are integers; b1, b2, b3 are rational numbers, c1, c2, c3 are irrational numbers This gives us practically nothing. This system of equations can definitely have a unique solution. So, only Statement i tells us that a unique solution is impossible. The question is "Which of the following statements if true would imply that the above system of equations does not have a unique solution?" ##### Hence the answer is "Statement i: $\frac{a_{1}}{a_{2}}\\$ = $\frac{b_{1}}{b_{2}}\\$ = $\frac{c_{1}}{c_{2}}\\$ ≠ $\frac{d_{1}}{d_{2}}\\$" Statement i is the correct answer. ###### Prepare for CAT 2022 with 2IIM's Daily Preparation Schedule ###### Know all about CAT Exam Syllabus and what to expect in CAT ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2022 Classroom Batches Starting Now! @Gopalapuram ###### Best CAT Coaching in Chennai Introductory offer of 5000/- Attend a Demo Class ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
Mobile: (91) 99626 48484
WhatsApp: WhatsApp Now
Email: prep@2iim.com