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Solved Examples

Question 1: Solve
2x + 3y = 5  and  5x - 2y = 3

Solution:
Given equations are
2x + 3y = 5                              ........................(1)

5x - 2y = 3                               ........................(2)

Step 1:

To eliminate x:
Multiply equation (1) by 5 and equation (2) by 2

=> 5(2x + 3y = 5)

=> 10x + 15y = 25                  ........................(3)

and 2(5x - 2y = 3)

=> 10x - 4y = 6                     ............................(4)

Step 2:

Subtract equation (4) from equation (3)

=> 10x + 15y - (10x - 4y) = 25 - 6  

=> 10x + 15y - 10x + 4y = 19

=> 19x = 19

Divide each side by 19

=> x = 1

Step 3:

Put x = 1 in equation (1)

=> 2 * 1 + 3y = 5  

=> 2 + 3y = 5

=> 3y = 5 - 2 = 3

Divide each side by 3

=> y = 1

Hence the solution to the system is (x, y) = (1, 1).

 

Question 2: A boat takes 20 hours to go 40 km downstream and 30 km upstream. Again the same boat takes 30 hours to go 20 km downstream and 25 km upstream. Find the speed of the boat and the current.
Solution:
Let speeds of the boat and the current be x km/hr and y km/hr.

Speed of the boat in downstream = (x + y) km/hr

Speed of the boat in upstream = (x - y) km/hr

Step 1:

According to problem:

$\frac{40}{x + y}$ + $\frac{30}{x - y}$ = 20                     .....................(1)

and
$\frac{20}{x + y}$ + $\frac{25}{x - y}$ = 30                    ..................... (2)

Step 2:

Put $\frac{1}{x + y}$ = u   and  $\frac{1}{x - y}$ = v

(1) => 40u + 30v = 20

or  4u + 3v = 2                                      ..........................(3)

and
(2) => 20u + 25v = 30

or 4u + 5v = 6                                      ............................(4)

Step 3:


Subtract (4) from (3)

=> 4u + 3v - (4u + 5v) = 2 - 6

=> 4u + 3v - 4u - 5v = -4

=> - 2v = - 4

=> v = 2, put in equation (3)

=> 4u + 3 * 2 = 2

=> 4u + 6 = 2

=> 4u = - 4

=> u = -1

Step 4:
 Now
 
$\frac{1}{x + y}$ = u  = -1

=> x + y = -1

or y = -1 - x             ......................(5)

and  $\frac{1}{x - y}$ = v = 2

=> 2x - 2y = 1, Put (5) in this equation

=> 2x - 2(-1 - x) = 1

=> 2x + 2 + 2x = 1

=> 4x = -1

=> x = $\frac{-1}{4}$, put in (5)

=> y = -1 - $\frac{-1}{4}$

=> y = -1 + $\frac{-1}{4}$

=> y = $\frac{-3}{4}$

[Since distance and speed can't be negative.]
Hence, speeds of the boat and the current be $\frac{1}{4}$ km/hr and $\frac{3}{4}$ km/hr respectively.