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1 Learning Objectives for Section 4.1 After this lesson, you should be able to solve systems of linear equations in two variables by graphing solve these systems using substitution solve these systems using elimination by addition solve applications of linear systems. Review: Systems of Linear Equations in Two Variables

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2 Consistent Systems Consistent system- a system of equations that has a solution. (the system has at least one point of intersection) one solution existsinfinitely many solutions exist

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3 Inconsistent Systems Inconsistent system- a system of equations that has no solutions. (the lines are parallel) no solution exists

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4 Independent and Dependent Equations Independent Equations- the equations graph different lines Dependent Equations- the equations graph the same line one solution existsno solution exists infinitely many solutions exist

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5 Special Cases Summary When 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 dependent The system is consistent.

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6 Special Cases Summary When 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.

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7 Supply and Demand The 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.

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8 Supply 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.

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9 Supply 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.

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10 Supply 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.

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11 Supply 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)

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12 Supply 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 where q is in thousands of cases and p represents the price per bulb in dollars. Find the equilibrium price and quantity.

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13 Supply 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 q Demand: p = -0.81q + 7.5

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14 Supply and Demand Example (continued)

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15 Supply and Demand Example Method II: Solve the system graphically Graph the two equations in the same coordinate system using a graphing calculator and find the ______________________ _________________________________________________. p = 1.04 q p = -0.81q y1= 1.04 x y2= -0.81x Notice x represents the quantity and y represents the price.

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16 Supply and Demand (Example continued) If we graph the two equations on a graphing calculator and find the intersection point, we see the graph below. Thus the equilibrium point is (7.854, 1.14). The equilibrium quantity is ____________ cases and the equilibrium price is $____________ per bulb. Supply Curve Demand Curve

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