Linear equation: word problem

Question

the sum of three numbers is 30. The rest number is one less than three times the second. the third number is five less than two times the second. Find each number and label respectively.

Answer 1

Let x = the second number. Then, (3x - 1) = the first number, and (2x - 5) = the third number

 

(3x - 1) + x + (2x - 5) = 30

6x - 6 = 30

6x = 36

x = 6

 

The second number is 6

 

The third number is 2x - 5 = 2 * 6 - 5 = 12 - 5 = 7

 

The first number is 3x - 1 = 3 * 6 - 1 = 18 - 1 = 17

Answer 2

Let the three numbers be x, y, and z.

 

Translate:

   "the sum of three numbers is 30"     means   x+y+z = 30     [eq1]

 

   "first number is one less than three times the second"    means

           x          =     -1       +      3       *             y                [eq2]

 

   "third number is five less than two times the second"   means

          z             =     -5       +     2     *              y                [eq3]

 

Note that values are given in terms of y, so let's eliminate them and keep y.

 

     x      +      y      +      z      =   30           [eq1]

 (3y-1)  +      y       +     z       =   30         [use eq1]

 (3y-1)  +      y       +  (2y-5)   =   30        [use eq3]

 

   6y - 6 = 30              [collect terms]

     6y = 36               [add 6 to both sides]

      y = 6           [divide both sides by 6]

 

Now, use the value of y to solve for x and z.

     x = -1 + 3(6)      [from eq2]

     x = 17

 

     z = -5 + 2(6)     [from eq3]

     z = 7

 

The numbers are:

     first = 17

     second = 6

     third = 7

 

Check:

  Is   17 + 6 + 7 = 30   ?

             30 = 30   ?yes