Math Olympiad Test: Linear Equations in Two Variables- 4 - Free MCQ

Two lines can have solution in two different cases,
Case 1: When they intersect at one point (unique solution)
Case 2: When the lines are coincident (infinite solutions)
Thus, if a pair of linear equations is consistent, then the lines will be intersecting or coincident.

Let the two numbers be x and y
Then, x + y = 64 ...(1)
Also, x – y = 22 ...(2)
or y – x = 22 ...(3)
(i) If x – y = 22, from (1) and (2), we have 2x = 86
⇒ x = 43 and y = 21
(ii) If y – x = 22, from (1) and (3), we have 2y = 86 ⇒ y = 43 and x = 21
∴ Numbers are 43 and 21.

We have, 36x + 24y = 702
and 24x + 36y = 558
Simplifying above equations, we get
6x + 4y = 117 ...(1) and 4x + 6y = 93 ...(2)
Multiplying eq. (1) by 3, eq. (2) by –2 and then adding, we get 18x + 12y – 8x – 12y = 351 – 186
⇒ 10x = 165 ⇒ x = 165/10 = 33/2 

We have,
  ......(1)
Also,
  ...(2)
Let 1/y = a and 1/x = b
So, (1) and (2) become
2a + 5b = 6 ...(3)
and, 4a – 5b = –3 ...(4)
Adding (3) and (4), we get
a = 1/2 and b = 1 [From (3)]
∴ x = 1 and y = 2

Let Arun has x books and Prabhat has y books
According to the question,
x − 3 = (1/2)(y + 3)⇒ 2x = y + 9 ... (1)
Also, x + 2 = y – 2 ⇒ x + 4 = y ... (2)
Solving (1) and (2), we get 
x = 13 and y = 17 ⇒ x + y = 30

For coincident lines, 
i.e., 

Let speed of boat in still water = x km/h
Speed of stream current = y km/h
According to question,

In these type of questions, make factor of 24 and 36 and choose the common values which satisfy the above equations.
24 = 2, 3, 4, 6, 8, 12
36 = 3, 4, 9, 12
Choose the common factor i.e. Put this value in equation (i)

x − y = 8
∴ x + y = 12
∴ x = 10, y = 2
Speed of the current,
y = 2 km/h.

Let x km/hr be the speed of truck starting from M and y km/hr be the speed of truck starting from N.
Case I : If they meet at P.


⇒ 10 + y = x ...(1)
Case II : If they meet at P.

For M, 
⇒ 70 = x + y ...(2)
Solving (1) and (2), we get
y = 30 km/hr and x = 40 km/hr

Let cost price of tea-set = ₹ x
and cost price of lemon-set = ₹ y
(i) Loss on tea-set = 
Gain on lemon-set = 
Also,  ...(1)
(ii) Gain on tea-set = 
Gain on lemon-set = 
Also,  ..(2)
Solving (1) and (2), we get ⇒ y = ₹ 80 and x = ₹ 100

Let the year in which Ram was born be x and the year in which Ram’s grandpa was born be y
Then, according to question, 
⇒ 2x – y = 2002 ...(1)
and x + y = 3854 ...(2)
Solving (1) and (2), we get ⇒ x = 1952
Thus in 2003, Ram’s age would be 2003 – 1952 = 51 yrs.