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SIMPLE WORD PROBLEMS RESULTING IN A SYSTEM OF EQUATIONS

• STEP 1: NAME YOUR UNKNOWNS
  Decide what piece(s) of information are UNKNOWN, and give name(s) to these things.
  Choose names that help you to remember what they represent!
  
  Let  n  be the number of night tickets (evening shows).
  Let  d  be the number of day tickets (matinee shows).
   
• STEP 2: WHAT CAN WE WRITE DOWN THAT IS TRUE?
  Re-read the word problem.
  The English words will translate into mathematical sentences involving your unknowns.
  You may need additional mathematical concepts in making your translation.
  
  

  ENGLISH WORDS	TRANSLATION INTO MATH	NOTES/CONVENTIONS
  "Antonio went to see a total of 12 movies"	n+d=12	NOTE: There are many real-number choices for  n  and  d  that make this equation true! Here are a few: 0 + 12 = 12 1.3 + 10.7 = 12 (-2) + 14 = 12 Of course, we want whole number solutions, and we also need something else to be true!
  "... and spent $86.00"	8n+6d=86	Each night movie costs $8.00, so  n  night movies cost  8n  dollars. Each day movie costs $6.00, so  d  day movies cost  6d  dollars. Both  8n  and  6d  have units of dollars. Also, the number  86  has units of dollars. It is important that you have the same units on both sides of the equal sign. Here, we have: dollars plus dollars is dollars. CONVENTION: Write  8n , not (say)  8.00n  or  $8n  or  $8.00n . NOTE: Convince yourself that there are also infinitely many real-number choices for  n  and  d  that make this equation true. We want choices for  n  and  d  that make BOTH equations true!

  
   
• STEP 3: CHOOSE A SIMPLEST EQUATION, AND SOLVE FOR ONE VARIABLE IN TERMS OF THE OTHER
  Remember that to solve for a variable means to get it all by itself, on one side of the equation,
  with none of that variable on the other side.
  Here, you're getting a  new name  for one of your variables that is helpful for finding the solution.
  
  Clearly, the equation   n+d=12   is simpler than   8n+6d=86 .
  We could solve the equation  n+d=12  for either  n  or  d ; hmmm... think I'll choose to solve for  n . (It doesn't matter!)
  Subtracting  d  from both sides, we get:   n=12-d .
   
• STEP 4: USE YOUR NEW NAME IN THE OTHER EQUATION
  Take your new name from the previous step, and substitute it into the remaining equation.
  This will give you an equation that has only one unknown.
  
  Substituting   n=12-d   into the equation   8n+6d=86   gives:   8(12-d )+6d=86  .
   
• STEP 5: SOLVE THE EQUATION FOR ONE UNKNOWN
  Solve the resulting equation in one variable.
  Be sure to write a nice, clean list of equivalent equations.
  
  

  8(12-d )+6d=86	original equation
  96-8d+ 6d=86	distributive law
  96-2d= 86	combine like terms
  -2d=- 10	subtract  96  from both sides
  d=5	divide both sides by  -2

  
   
• STEP 6: USE THE KNOWN VARIABLE TO FIND THE REMAINING VARIABLE
  Go back to the simplest equation, substitute in your new information,
  and solve for the remaining variable.
  
  Make sure you understand the logic being used:
  If both   n+d=12   and   8n+6d=86   are true, then  d  must equal  5 .
  
  Substitute   d=5   into the simple equation   n+d=12   and solve:
  
  

  n+d=12	the simple equation
  n+5=12	substitute in the known information
  n=7	subtract  5  from both sides

  
   
• STEP 7: CHECK, and REPORT YOUR SOLUTIONS
  Check that the numbers you've found make both of the equations true.
  Then, report your solution(s) using a complete English sentence.
  
  

  EQUATIONS	CHECK	TRUE?
  n+d=12	7+5= ?12	Yes!
  8n+6d=86	8(7)+ 6(5)= ?86	Yes! (Feel free to use your calculator.)

  
  The original problem asked how many night movies Antonio attended, so here's what you'd report as your answer:
  
  Antonio attended  7  night movies.

THE GOOD NEWS!

Let  n= # night tickets. Let  d= # day tickets. n+d=12 8n+6d=86 n=12-d 8(12-d )+6d=86 96-8d+ 6d=86 96-2d= 86 -2d=- 10 d=5     (circle this) n+5=12 n=7     (circle this) 7+5= ?12     8(7)+ 6(5)= ?86     Antonio attended  7  night movies.