When solving equations with two variables, we need two facts.  There are two unknowns so we need two conditions in order to solve.

Suppose we have A + B = 5.  We know that if A = 4, then B = 1 but we can't know either value just from A + B = 5.  If we were to graph that equation it would be a line and that would show an infinite number of combinations.  If we add another condition such as A - B = 4 and graph that then we have another line that crosses the first one.  That point is where both A + B = 5 and A - B = 4 are true.  Of course, this example is easy enough that we don't really need to have algebra training to solve it. 

O f course in the previous problem there seems to be no real purpose.  So, let's think of it like this riddle: What two numbers have a sum of 5 and a difference of 4?

A kind of classic algebra problem would be one where a car leaves at 12:00 traveling at a steady 30 mph.  A second car leaves at 2:00 traveling at a steady 40 mph.  At what time would they have traveled the same distance (or the second car catches the first)?

We can do this in a variety of ways.  Some are algebraic and some are graphical.  Many of the ways are made easier by the graphing calculator.  We often do this often by thinking about the graphs.  You might want to also see solving linear equations.

One thing you may note about studying mathematics is that even the simplest of ideas can quickly become complex.  Example: a common formula used by most everyone is d = rt  where distance d is calculated by multiplying the rate of travel by the time.  Simple?  Yes, but when was the last time you traveled by car or by bicycle or even walking where you maintained a constant speed?  We almost always vary our speeds.  Also, when traveling in the real world, we have external factors that change things for us.  For instance, pilots learn that their plane's speed is altered by the winds.  A headwind or a tailwind will change the time of travel and can be an important factor when figuring fuel consumption. 

But back to the simple:

If two numbers have a sum (add) of 5 and a difference (subtract) of 4 we can write it like this:  A + B = 5 and also A - B = 4 .

So for the speeds problem we can also use equations to set it up.   We know that d = rt  so the first car has the equation d = 30t where t is the time after 12:00 and the second car has the equation d = 40(t + 2) where t is the time after 12:00.  (We add 2 because it left 2 hours later.)  

Now we need a method for solving.

There are two standard ways:  substitution and elimination