Question

the difference between two numbers is 8.If 2 is added to the bigger number the result will be 3times the smaller number. Find the numbers.

Answer 1

Question: The difference between two numbers is 8. If 2 is added to the bigger number, the result will be 3 times the smaller number. Find the numbers.
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Solution:

Let  - the larger number be $x$.
   &    the smaller number be $y$.

From the given information, we form two equations:

                            $x-y=8$       ... (1.)
                            $x+2=3y$     ... (2.)

We must keep the x's with the x's, the y's with the y's, and the integers with the integers. All must be kept at specific places.

   The arrangement be: X (sign) Y = INTEGER
               
Since the elements in second equation are not arranged in the proper order, it (referring to the second equation) is to be modified.


$x+2=3y$ 
is equal to          $x-3y=-2$


Next, we subtract the second equation from the first equation. This way, y can be isolated and the x's eliminated.

                                        $x-y=8$
                               −       $x-3y=-2$       
                        ------------------------------------------------- 
Resulting equation:    
                                   $0(x)+2y=10 \\ 2y=10$
                                ∴  $y=5$  (ans.)

As we have solved for y, we can now find x with ease by inserting the value of y in any of the equations. Afterwards, we solve for x.        
 
 I here choose the first equation. Either of the equations can be used to solve for x. It doesn't matter which equation we use to solve. It simply depends upon the preference of the one who is solving the problem.    
   
                                      $x-y=8$      
       Insert 5 (the value of y) in the place of the variable y:
                                      $x-5=8$
                                    
                             =>     $x=8+5$
                             =>     $x=13$ (ans.)

Answer: Therefore, the numbers are 13 and 5.

Answer 2

x=larger number; y=smaller number . x-y=8 x=y+8 Use this to substitute for x. . x+2=3y Substitute for x from above. (y+8)+2=3y y+10=3y Subtract y from each side. 10=2y Divide each side by 2. , 5=y ANSWER 1: the smaller number is 5. . ,x=y+8=5+8=13 ANSWER 2: The larger number is 13. . CHECK: x+2=3y 13+2=3(5) 15=15