The sum of two numbers is 26. The second is 2 less than 3 times the first. What are the numbers?

Question

I have always been good at math but for some reason have always struggled with word problems.  Hopefully this helps.

Answer 1

The key to solving word (story) problems is to translate the words and phrases into concise, precise math expressions (variables and operations).

 

First, assign variables:

    Let F = first number

         S = second number

 

Then translate:

   "sum of two numbers is 26"    means    F + S = 26         (note:  "sum" means add)

   "second is 2 less than 3 times the first"   means  S = 3F -2     (note: 'is" means =)

   "What are the numbers?"    means    Report F, S

 

Now, the easy math   (let's use elimination method):

      F + S  = 26

   -3F + S  = -2       (from S=3F-2, subtracting 3F from both sides)

  -----------------

    4F         = 28        (subtract equations)

       F = 7              (divide both sides by 2)

 

Again, let's eliminate:

       F + S = 26    (from first equation)

       F       = 7

    --------------   (subtract equations)

             S = 19

 

Checking (very important):

   Is  7 + 19 = 26  ?     yes

   Is  19 = 3*7 - 2  ?

        19 = 19   ?   yes !

 

Answer 2

The best way to do this in algebra is to break it down into 2 equations:

 

x + y = 26 (where x is the first and y is the second #)

 

y = 3x - 2

 

Then, you can substitute the second into the first equation (insert "3x - 2" everywhere you see y):

 

x + (3x - 2) = 26

 

4x - 2 = 26   Combine like terms (the parentheses were just for demonstration purposes)

4x      = 28   Add +2 to both sides

 x       =  7    Divide both sides by 4

 

Now, to get y, plug 7 back in one of the equations and solve.

7 + y = 26

      y = 19   Subtract 4 from both sides