Presentation on theme: "Learning Objectives for Section 4.1"— Presentation transcript:
1Learning Objectives for Section 4.1 MAT 103 SP09Learning Objectives for Section 4.1Review: Systems of Linear Equations in Two VariablesAfter this lesson, you should be able tosolve systems of linear equations in two variables by graphingsolve these systems using substitutionsolve these systems using elimination by additionsolve applications of linear systems.
2Consistent SystemsConsistent system- a system of equations that has a solution.(the system has at least one point of intersection)one solution existsinfinitely many solutions exist
3Inconsistent SystemsInconsistent system- a system of equations that has no solutions.(the lines are parallel)no solution exists
4Independent and Dependent Equations Independent Equations- the equations graph different linesone solution existsno solution existsDependent Equations- the equations graph the same lineinfinitely many solutions exist
5Special Cases SummaryWhen solving a system of two linear equations in two variables algebraically:If an identity is obtained, such as 0 = 0, then the system has an infinite # of solutions.The equations are dependentThe system is consistent.
6Special Cases SummaryWhen solving a system of two linear equations in two variables algebraically:If a contradiction is obtained, such as 0 = 7, then the system has no solution.The system is inconsistent.The equations are independent.
7Supply and DemandThe quantity of a product that people are willing to buy during some period of time is related to its price.Generally, higher their price, less their demand; and lower their price, then greater their demand.
8Supply and Demand (continued) Similarly, the quantity of a product that a supplier is willing to sell during some period of time is also related to the price.Generally, a supplier will be willing to supply more of a product at higher prices and less of a product at lower prices.
9Supply and Demand (continued) The simplest supply and demand model is a linear model where the graphs of a demand equation and a supply equation are straight lines.
10Supply and Demand (continued) In supply and demand problems we are often interested in finding the price at which supply will equal demand.This is called the equilibrium price, and the quantity sold at that price is called the equilibrium quantity.
11Supply and Demand (continued) If we graph the the supply equation and the demand equation on the same axis, the point where the two lines intersect is called the EQUILIBRIUM POINT.Its horizontal coordinate is the value of the equilibrium quantity (q).Its vertical coordinate is the value of the equilibrium price (p).(q, p)
12Supply and Demand Example Example: Suppose that the supply equation for long-life light bulbs is given by(supply) p = 1.04 q ,and that the demand equation for the bulbs is(demand) p = -0.81q + 7.5where q is in thousands of cases and p represents the price per bulb in dollars.Find the equilibrium price and quantity.
13Supply and Demand Example Method I: Solve the system algebraically We want to find the price at which the supply is equal to the demand.We can do this by __________________________________________________________________________________.Supply: p = 1.04 qDemand: p = -0.81q + 7.5
15Supply and Demand Example Method II: Solve the system graphicallyGraph the two equations in the same coordinate system using a graphing calculator and find the _______________________________________________________________________.p = 1.04 qp = -0.81q + 7.5y1= 1.04 xy2= -0.81x + 7.5Notice x represents the quantity and y represents the price.
16Supply and Demand (Example continued) If we graph the two equations on a graphing calculator and find the intersection point, we see the graph below.Demand CurveSupply CurveThus the equilibrium point is (7.854, 1.14).The equilibrium quantity is ____________ cases and the equilibrium price is $____________ per bulb.