4-1 Systems of Equations (two variables)
Consider the following system of equations:
y=x+2
y = -x + 4
Our job: find a point (pair (x,y)) that satisfies both equations
If we graph these equations, we get:
4
y
y=x+2
y = -x + 4
x
4
 describe the point that satisfies both equations
 what are its coordinates?
 does that point, in fact, satisfy both equations?
graphical method (just shown):
 can be an effective way, using a graphing calculator
 we will concentrate on systematic, algebraic methods
4-1
p. 1
Solving systems by elimination
Example:
1
x+y=4
2
x- y=1
Strategy: “add” the two equations in order to "eliminate" one of
the variables
+
1
x+y=4
2
x- y=1
_____________________
3
2x
=5
You now have one equation with one unknown.
Solve it for x, substitute in 1 or 2 and solve for y.
Example:
1
3x + 3y = 15
2
2x + 6y = 22
Mere adding won't eliminate. We multiply each equation by an
appropriate constant so adding will eliminate (either x or y):
1  2:
6x + 6y = 30
2 -3:
-6x - 18y = -66
___________________________________
-12y = -36
y=3
4-1
etc.
p. 2
FUNNY THINGS THAT CAN HAPPEN, AND HOW TO COPE
Funny thing #1:
How to cope:
Example:
you get an equation that is never true, e.g.
10 = 0
STOP! and write the answer "No solution"
 system consists of two parallel lines (they
never cross)
 called an inconsistent system
x - y = 10
-x + y = 5
_________________________
0 + 0 = 15
Ans: inconsistent; no solution
Funny thing #2:
How to cope:
Example:
you get an equation that is always true, e.g.
10 = 10
STOP! and write as shown in example
 system consists of two identical lines
(every point of the first is a point on the
second)
 all (infinitely many) points on either line
are solutions
 called a dependent system
x - y = 10
-x + y = -10
_______________________
0+0=0
Ans: dependent system
all points on the line x - y = 10
4-1
p. 3
An application: supply and demand
Pokey, Inc. makes Pokemon cards. Two variables are
involved:
 q: the quantity of cards to be manufactured
 p: the price per card to be charged the consumers
There are two governing equations that embody what is
known as the "law of supply and demand":
 price-demand equation
 q = -1000p + 3000
 describes the effect of price on demand
 price-supply equation
 q = 1200p - 800
 describes the effect of price on supply
Note:
Our book would write these equations (equivalently) as
p = -(1/1000)q + 3, and p = (1/1200)q + 2/3, that is,
thinking of p as a function of q, or, q as the independent
variable and p as the dependent variable.
I think things are easier to explain by taking p to be the
independent variable.
4-1
p. 4
Supply and demand equations graphed
demand equation: q = -1000p + 3000 (here, q is demand)
supply equation: q = 1200p – 800
(here, q is supply)
demand equation has negative slope . . .
 as the price increases, demand will drop
 consumer market force
supply equation has positive slope . . .
 as the price increases, the supply will increase
 supplier market force
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p. 5
Market forces create a tug-of-war in the market place, as
follows:
If price (p) = $1:
demand (q) = 2000 cards
supply (q) = 400 cards
Demand exceeds supply, so the price will go up
Suppose price rises to $2:
demand = 1000
supply = 1600
Now supply exceeds demand, and price will be forced
down.
Here's the picture:
At a price of $1, market forces send the price up 
At $2, market forces send the price down

4-1
p. 6
Will the market inevitably continue to move up and down?
Or is there a price at which the kids will buy exactly the the
quantity being produced, achieving an equilibrium
price/quantity?
Mathematically, the equilibrium values will be found by
solving the supply and demand equations simultaneously;
doing so, we get p = $1.72, and q = 1280. So if Pokey
manufactures 1280 cards and sells them for $1.72,
everybody will be satisfied (!!).
The troubles:
 the assumption that these relationships are linear, and
(assuming that they are) . . .
 determining the appropriate equations for the given
market
4-1
p. 7