CHAPTER 1: FUNCTIONS, GRAPHS, AND MODELS; LINEAR FUNCTIONS Section 1.7: Systems of Linear Equations in Two Variables 1

SECTION 1.7: Systems of Linear Equations in Two Variables The break even point is a term used to represent the point at which the Revenue = Cost. (R(x) = C(x)) If we know R(x) and C(x), we can find the point at which the functions ‘ meet ’ graphically. Graph both functions and identify the point of intersection – this marks the break even point. 2

SECTION 1.7: Systems of Linear Equations in Two Variables Suppose a company has its total revenue, in dollars, for a product given by R(x) = 5585x and its total cost in dollars is given by C(x) = 61, x where x is the number of thousands of tons of the product that is produced and sold each year. Determine the break even point and the corresponding values for R and C. Interpret. 3

SECTION 1.7: Systems of Linear Equations in Two Variables Let Y 1 = R(x) and Y 2 = C(x). Graph both functions – be sure to use a window that allows you to locate the intersection. Find the point of intersection. 2 nd TRACE (CALC) 5: intersect Follow First and Second Curve Prompts (use the up and down arrows to move from curve to curve) Move the cursor to an approximate location for the guess. 4

SECTION 1.7: Systems of Linear Equations in Two Variables Will two lines always intersect? If 2 equations graph the same line, we say the system is a dependent system. There are an infinite number of solutions for such as system. If 2 equations graph parallel lines, we say the system is inconsistent. There is No solution for such a system. If 2 equations graph with an intersection, we say the system is consistent. There is one unique solution for such a system. 5

SECTION 1.7: Systems of Linear Equations in Two Variables Solution by Substitution Solve one of the equations for one of the variables in terms of the other variable. Substitute the expression from step 1 into the other equation to give an equation in one variable. Solve the linear equation for the variable. Substitute this solution into the equation from step 1 or into one of the original equations and solve this equation for the second variable. Check the solution in both original equations or check graphically. 6

SECTION 1.7: Systems of Linear Equations in Two Variables Solve the system below by substitution: 7

SECTION 1.7: Systems of Linear Equations in Two Variables Solution by Elimination If necessary, multiply one or both equations by a nonzero number that will make the coefficients of one of the variables in the equations equal, except perhaps for the sign. Add or subtract the equations to eliminate one of the variables. Solve for the variable in the resulting equation. Substitute the solution from step 3 into one of the original equations and solve for the second variable. Check the solutions in the remaining original equation, or graphically. 8

SECTION 1.7: Systems of Linear Equations in Two Variables Solve the system below by elimination: 9

SECTION 1.7: Systems of Linear Equations in Two Variables Market equilibrium Demand is the quantity of a product demanded by consumers. Supply is the quantity of a product supplied. Both demand and supply are related to the price. Equilibrium price is the price at which the number of units demanded equals the number of units supplied. We can also refer to this as market equilibrium. 10

SECTION 1.7: Systems of Linear Equations in Two Variables Suppose the daily demand for a product is given by p = 200 – 2q, where q is the number of units demanded and p is the price per unit dollars. The daily supply is given by p = q, where q is the number of units supplied and p is the price in dollars. If the price is $140, how many units are supplied and how many are demanded? Does this price result in a surplus or shortfall? What price gives market equilibrium? 11

SECTION 1.7: Systems of Linear Equations in Two Variables A nurse has two solutions that contain different concentrations of certain medication. One is a 12% concentration, and the other is an 8% concentration. How many cubic centimeters (cc) of each should she mix together to obtain 20cc of a 9% solution? 12

SECTION 1.7: Systems of Linear Equations in Two Variables An investor has $300,000 to invest, part at 12% and the remainder in a less risky investment at 7%. If her investment goal is to have an annual income of $27,000, how much should she put in each investment? 13

SECTION 1.7: Systems of Linear Equations in Two Variables Inconsistent and Dependent Systems Use elimination to solve each system, if possible. 14

SECTION 1.7: Systems of Linear Equations in Two Variables Homework: pp every other odd, 33, 37, 41, 45, 49, 53, 57 15