Transcript Section 1.4 Intersection of Straight Lines

Section 1.4
Intersection of Straight Lines
Intersection Point of Two Lines
Given the two lines
L1 : y  m1 x  b1
L2 : y  m2 x  b2
m1 ,m2, b1, and b2 are constants
Find a point (x, y) that satisfies both equations.
L1
L2
Solve the system consisting of
y  m1 x  b1 and y  m2 x  b2
Ex. Find the intersection point of the following
pairs of lines:
y  4x  7
y  2 x  17
4x  7  2x 17
6 x  24
x4
Notice both are in
Slope-Intercept Form
Substitute in for y
Solve for x
y  4x  7
 4(4)  7  9
Find y
Intersection point: (4, 9)
Break-Even Analysis
The break-even level of operation is the level of
production that results in no profit and no loss.
Profit = Revenue – Cost = 0
Revenue = Cost
break-even
point
Dollars
Revenue
profit
loss
Cost
Units
Ex. A shirt producer has a fixed monthly cost of $3600.
If each shirt has a cost of $3 and sells for $12 find the
break-even point.
If x is the number of shirts produced and sold
Cost: C(x) = 3x + 3600
Revenue: R(x) = 12x
R( x)  C ( x)
12 x  3x  3600
x  400
R (400)  4800
At 400 units the break-even revenue is $4800
Market Equilibrium
Market Equilibrium occurs when the quantity
produced is equal to the quantity demanded.
price
supply
curve
demand
curve
x units
Equilibrium Point
Ex. The maker of a plastic container has determined
that the demand for its product is 400 units if the unit
price is $3 and 900 units if the unit price is $2.50. The
manufacturer will not supply any containers for less
than $1 but for each $0.30 increase in unit price above
the $1, the manufacturer will market an additional 200
units. Both the supply and demand functions are linear.
Let p be the price in dollars, x be in units of 100 and
find:
a. The demand function
b. The supply function
c. The equilibrium price and quantity
a. The demand function
3  2.5
 x, p  :  4,3 and 9, 2.5 ; m  4  9  0.1
p  3  0.1 x  4
p  0.1x  3.4
b. The supply function
1  1.3
 x, p  :  0,1 and  2,1.3 ; m 
 0.15
02
p  0.15 x  1
c. The equilibrium price and quantity
Solve p  0.1x  3.4 and p  0.15 x  1
simultaneously.
0.1x  3.4  0.15x 1
0.25x  2.4
x  9.6
p  0.15(9.6)  1  2.44
The equilibrium quantity is 960 units at a price
of $2.44 per unit.