Algebraic Translations
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In the lesson on Algebraic Representations we looked at writing mathematical expressions.
We are ready now to write mathematical equations.

Expressions:
 10,   2 + y,   z,   7xy, 
    5x - 6,   2(a + 3),
         x2 + 4x + 4
   "Expressions" do NOT have an equal sign.
Equations:               
2 + y = 10
    5x - 6 = 41
       x2 + 4x + 4 = 0
        "Equations" have an EQUAL sign.


"Expressions" and "equations" refer to two different entities in mathematics.
If you are dealing with an EQUAL SIGN, you are dealing with an "equation".

EQUAtion ↔ EQUAL sign
 An equation is a sentence where two algebraic expressions are equal.
 

Examples

Translate the following problems into equations:
HINT:  If you are having trouble determining what the equation might be, make a numerical problem to see how the numbers in the problem are related to one another.

1. Cost for cable TV service in a certain city is $45 for the installation and $24 a month for service.
Number problem:  for 3 months the cost would be:  Cost = $24 • 3 + $45.  Replace 3 with m.

Let c = cost, m = number of months
c = 24m + 45
 
2. If golf balls cost $1.25 and putters cost $79, how many of each can the golf team purchase for $150?
Number problem:  if I purchase 2 golf balls and 3 putters the cost will be:   Cost = 1.25•2 + 79•3
Replace 2 with g (number of golf balls), 3 with  p (number of putters), and Cost with $150.

Let g = number of golf ball,
 p = number of putters
1.25g + 79p = 150
 
3. H2O means that water contains two hydrogen atoms and one oxygen atom.
Number problem:  In water, if there is one oxygen atom, there are 2 hydrogen atoms; if there are two oxygen atoms, there are 4 hydrogen atoms and so on.  The number of hydrogen atoms is twice the number of oxygen atoms.

Let O = number of oxygen atoms,
 H = number of hydrogen atoms
H = 2O

4.

 A cookie recipe calls for twice as many chocolate chips as walnuts.
Number problem:  if a recipe calls for 10 walnuts, it calls for 20 chocolate chips.  The number of chocolate chips is twice the number of walnuts.

Let c = number of chips,
 w = number of walnuts
c = 2w

5.

 What is the total weight of a filled hot tub, if the tub weighs 350 pounds and the water weighs 10 pounds per gallon?
Number problem:  if it takes 100 gallons to fill the hot tub, the total weight will be:
Total = 10 • 100 + 350  Replace 100 with g, for the number of gallons..

Let t = total weight,
g = number of gallons
t = 10g + 350