The sum of two numbers is 32. the larger number is one less than twice the smaller number. find the numbers.

The numbers are:  "11" and "21" .

Explanation:

x + y = 32 ;     Note:  y  is greater than "x";    

y = 2x − 1 ; 

We have:  " x + y = 32" ;

Plug in "(2x − 1)" for "y" ;

    →  x + (2x − 1) = 32 ; 

    →   x + 1(2x − 1) = 32 ;
 
    →  x + 2x − 1 =  32 ;

    → 1x + 2x − 1 =  32 ;

    →  3x − 1 = 32 ;

Add "1" to each side of the equation ; 

   →  3x − 1 + 1 = 32 + 1 ; 

   →  3x = 33 ;

Divide each side of the equation by "3" ; 
 to isolate "x" on one side of the equation; & to solve for "x" ; 

           3x / 3 = 33 / 3  ;

             x = 11 ; 

Now, let us solve for "y" ;

Method 1)

We have:  "x + y = 32" ; 

We have "x = 11 " ;  

Plug in "11" for "x" into the equation; to solve for "y" ; 

     "x + y = 11 + y = 32 " ; 

→ y + 11 = 32 ;

Subtract "11" from each side of the equation; 

→ y + 11 − 11 = 32 − 11 ;

  →  y = 21 ; 

So; the numbers are:  "11" and "21" .

Method 2)  (and we can do this anyways to confirm our answer):

When we have:  "x = 11 " ;  Let us solve for "y" ; 

  We have the equation:  "y = 2x − 1 " ;

Plug in the value, "11", for "x", in this equation; to solve for "y" ; 

     → y = 2(11) − 1 ;

     → y = 22 − 1 ; 

     → y = 21 


 So, our answers are:  "11" and "21" .

Note:  x + y = ? 32 ? ;   → "11 + 21 = ? 32 ? " ; Yes!

Note:  "y" is the larger number;  y > x ? ;  →  "21 > 11 " ?  Yes!

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