Many word problems, upon translation, result in two equations involving two variables (two ‘unknowns’).
In mathematics, a collection of more than one equation being studied together is called a system of equations.
The systems in this section are fairly simple,
and can be solved by substituting information from one equation
into the other.
The procedure is illustrated with the following example:
Antonio loves to go to the movies. He goes both at night and during the day. The cost of a matinee is \$6.00. The cost of an evening show is \$8.00. If Antonio went to see a total of $\,12\,$ movies and spent \$86.00, how many night movies did he attend? 
ENGLISH WORDS  TRANSLATION INTO MATH  NOTES/CONVENTIONS 
“Antonio went to see a total of 12 movies”  $n+d = 12$ 
NOTE:
There are many realnumber choices for $\,n\,$
and $\,d\,$ that make this equation true.
Here are a few:
$0 + 12 = 12$
$1 + 11 = 12$
$1.3 + 10.7 = 12$
$(2) + 14 = 12$
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's 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 realnumber choices for $\,n\,$ and $\,d\,$ that make this equation true. We want a choice for $\,n\,$ and a choice for $\,d\,$ that make BOTH equations true at the same time. 
$8(12  d) + 6d = 86$  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$\,2\,$ 
$n+d = 12$  the simple equation 
$n + 5 = 12$  substitute in the known information 
$n = 7$  subtract $\,5\,$ from both sides 
EQUATIONS  CHECK  TRUE? 
$n + d = 12$  $7 + 5 \,\,\overset{\text{?}}{ = }\,\, 12$  Yes! 
$8n + 6d = 86$  $8(7) + 6(5) \,\,\overset{\text{?}}{ = }\,\, 86$ 
Yes! (Feel free to use your calculator.) 
Antonio loves to go to the movies. He goes both at night and during the day. The cost of a matinee is \$6.00. The cost of an evening show is \$8.00. If Antonio went to see a total of $\,12\,$ movies and spent \$86.00, how many night movies did he attend? 
Let $\,n = \text{# night tickets}\,$. Let $\,d = \text{# day tickets}\,$.
$8(7) + 6(5) \,\,\overset{\text{?}}{=}\,\,86$ ☺ Antonio attended 7 night movies. 
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
