Preliminaries:
Solving ax + b = c for x when a is not zero:
1) Subtract b from both sides. 2) Divide both sides by a.
Example 1. Solve the equation 4x+3=11 for x.
Subtract 3 from both sides to get 4x = 8.
Divide by 4 to get x = 2.
Example 2. Solve the equation -3x + 2 = 7 for x.
 3x  2  7
 3x  5
x 
5
 
3
5
3
Solving ax + b = cx + d for x. Subtract cx from both sides to get an equation like the ones
above.
Example 3. Solve the equation 2x + 3 = 4x - 5 for x.
2x  3  4x  5
2 x  4 x  5  3
(2  4) x  8
 2 x  8
x 
 8
 2
 4
Lines and Applications:
The slope intercept form of a line is y = mx+b. The point (0, b) is the y-intercept and m is
the slope.
If m is positive, the line rises as x moves to the right. If m is negative, the line falls as x
moves to the right.
Definition of slope between two points with different x coordinates:
(x
,y
),
P
x
,y
1
1
1
2(
2
2)is
The slope between the two points P
y y
rise
1
m 2

x2x
run
1
If the points have the same x coordinate the slope is undefined.
When a point and the slope of the line are given use the
(xx
1) to find the equation.
Point-slope form of a line: yy1 m
Example 1. Find the slope intercept form of the line through the point (2, 7) with slope 3.
y  7  3( x  2)
y  7  3x  6
y  3x 1
Example 2. Find the slope intercept form of the line through the two points (2, 6) and
(-2, 3).
6

3
3
3
3
3 3
9
m
 y

6

(
x

2
)y

6

x

y

x

2

(

2
)
4
4
4
2 4
2
Applications:
Linear Depreciation:
When a valuable loses value by the same amount each year, we say it depreciates
linearly.
Example 1. A new car sold for $24000 in 2004. It depreciated linearly to a value of
$13500 in 2007. Find the value of the car in terms of the number of years past 2004.
Linear Cost-Revenue-Profit. We assume x is the quantity produced and sold for a certain
product. The cost function is the cost to the manufacturer of producing x units. The
revenue is the amount of the receipts from selling x units and is the selling price times x.
Profit = revenue - cost=money taken in - money spent
Example 2:
A small manufacturer estimates his rent, machinery and other fixed costs total $1360.
each unit costs an additional $5 to produce and sells for $9.
a) Find the cost, revenue and profit functions.
b) Find the break even quantity, the quantity at which profit is 0 and revenue is equal to
cost.
c) Graph these three functions and label the y-intercepts, the break even point and the xintercept of the profit function.
Linear Supply and Demand:
The supply line describes the price at which the suppliers are willing to sell x units and
always has a positive slope.
The demand line describes the price at which consumers will buy x units and always has
a negative slope.
Example 3:Suppliers of a certain product will sopply none at a price of $30 or lower.
They will supply 100 at a price of $64. consumers will purchase 500 when the price is
$30. For each increase of $4 in the price, the quantity sold decreases by 25.
Note: The last sentence describes the slope of the demand equation. You can either create
a new point from that information or directly compute the slope.
a) Find the linear supply equation.
b) Find the linear demand equation.
c) Find the equilibrium quantity and price. This is the point where the supply and demand
lines intersect.