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Graphically Solving Systems of Two Equations (Page 1 of 24) 3.1 Graphically Solving Systems of Two Equations Definitions The plot of all points that satisfy an equation forms the graph of the equation. A system of linear equations is two or more linear equations considered together. The system in example 1 is referred to as a system of two linear equations in two variables. An ordered pair (a, b) that satisfies all equations in a system of equations is a solution to the system of equations. The set of all solutions to a system of equations makes up the solution set to the system. Example 1 Consider the system of linear equations: y 2x 1 y 3x 6 1. The sketch the graph of each equation is shown. What is the point of intersection in the two graphs? 2. Is there a point that satisfies both equations? 3. What is the relationship between the point(s) of intersection of the two lines and the solution to the system of equations. QuickTime™ and a TIFF (Uncompressed) decompressor are needed to see this picture. Graphically Solving Systems of Two Equations (Page 2 of 24) Example 2 Solve the system of equations by graphing. y 2x 4 y x 1 y 5 -5 5 x -5 Example 3 Graph the system of equations. Use your calculators to estimate the solution to the nearest hundredth. 1 5 5 x y y 2 4 2 y 3x 4 5 -5 5 -5 x Graphically Solving Systems of Two Equations (Page 3 of 24) Example 4 Solve an Inconsistent System Solve the system of equations by graphing. y y 2x 1 y 2x 5 5 -5 5 x -5 Distinct Parallel lines form an Inconsistent System Distinct lines with the same slope are parallel and do not intersect. In this case the solution set is empty and the system of equations is called an inconsistent system. Example 5 Solve a Dependent System Solve the system of equations by graphing. y 2x 4 6x 3y 12 y 5 -5 5 -5 Non-Distinct lines form a Dependent System Two lines with the same slope and same y-intercept are not distinct and have infinitely many points of intersection. In this case the solution set is every point on the line and the system of equations is called a dependent system. x Graphically Solving Systems of Two Equations (Page 4 of 24) The Solution Set to a Linear System of Two Equations in Two Variables has Three Possible Outcomes . . . 1. One Solution See Figure 1. The lines intersect at one point and the solution set contains only one point. A system of linear equations with at least one solution is called a consistent system. The solution to the system is expressed as an ordered pair (a, b) . y x Figure 1 A Consistent System with one solution. y 2. No Solutions See Figure 2. The lines are distinct and parallel, and never intersect. Therefore, there is no solution. A linear system whose solution set is empty is called an inconsistent system. 3. Infinite Number of Solutions See Figure 3. The lines lie on top of each other, are not distinct, and have an infinite number of points of intersection. Therefore, the system of equations has infinite solutions. This is called a dependent system. x Figure 2 An Inconsistent System has no solutions. y x Figure 3 A Dependent System has an infinite number of solutions. Graphically Solving Systems of Two Equations (Page 5 of 24) Example 6 Some values for functions f and g are given in the table. 1. What happens to the value of f each time x is increased by one? What is the slope of the linear function f? What is the y-intercept of f? x 0 1 2 3 4 5 6 7 f (x) 30 27 24 21 18 15 12 9 Find the linear function f. 2. What happens to the value of g each time x is increased by one? What is the slope of the linear function g? What is the y-intercept of g? Find the linear function g. 3. Use the Intersect function on your calculator to find the solution to the system of equations given by f and g. g(x) 2 7 12 17 22 27 32 37 Graphically Solving Systems of Two Equations (Page 6 of 24) Example 7 The life expectancies for men and women in the United States are given in the table. Let M (t) and W (t) represent the life expectancy of men and women, respectively, for t years since 1980. 1. Verify the regression equations are W (t) 0.098t 77.59 and M (t) 0.192t 69.99 . U.S. Life Expectancies of Women and Men Year of Women Men Birth (years) (years) 1980 77.4 70.0 1985 78.2 71.1 1990 78.8 71.8 1995 78.9 72.5 2000 79.5 74.1 2. Find the point of intersection. Explain its meaning in this application. 3. What is the slope of each equation? What does it mean in the context of this application? y Example 8 Write a system of equations that has (2, 3) as its only solution. 5 -5 5 -5 x Substitution and Elimination Methods (Page 7 of 24) 3.2 Solving a System of Equations by the Substitution and Elimination Methods What is the solution to the system of equations shown? Notice how difficult it is to read an accurate solution when the components of the ordered pair are non-integer values. y 5 In section 3.1 we solved a system of -5 5 two equations with two variables by graphing the two equations and reading the point(s) of intersection as -5 the solution. In this section we will learn two algebraic (symbolic) methods to solve a system of two equations with two variables: The Substitution Method and the Elimination Method. The Substitution Method 1. Solve one of the equations for one of the variables. 2. Substitute the equation from step 1 into the other equation and solve for the only remaining variable. 3. Substitute the result from step 2 into the equation from step 1 to find the value of the other variable. 4. Write the solution as an ordered pair. Example 1 Solve using substitution x y 1 3x 2y 13 QuickTime™ and a Photo - JPEG decompressor are needed to see t his picture. QuickTime™ and a Photo - JPEG decompressor are needed to see t his picture. Substitution and Elimination Methods (Page 8 of 24) Example 2 Solve using substitution y 11 3x y 2x 1 QuickTime™ and a Photo - JPEG decompressor are needed to see t his picture. QuickTime™ and a Photo - JPEG decompressor are needed to see t his picture. The Elimination Method 1. Rewrite one or both equations so that the coefficients on either variable are opposites. 2. Add the two equations from step 1 together, eliminating one of the variables. Solve for the value of the only variable present. 3. Use the result from step 2 and any equation to solve for the value of the remaining unknown variable. 4. Write the solution as an ordered pair. Example 3 Solve using elimination 4x 5y 3 3x 5y 11 Substitution and Elimination Methods (Page 9 of 24) Example 4 Solve using elimination Example 5 Solve using elimination 3x 2y 18 6x 5y 9 4x 3y 3 5x 2y 25 Substitution and Elimination Methods (Page 10 of 24) Example 6 Solve substitution or elimination. y 2x 1 y 2x 3 Identifying an Inconsistent System of Two Equations If the result of applying substitution or elimination is a false statement (such as 1 3 ), then the system is inconsistent and the solution set is empty. This occurs when the equations are two distinct lines with the same slope (and different y-intercepts). Example 7 Solve substitution or elimination. y 2x 1 4y 8x 4 Identifying a Dependent System of Two Equations If the result of applying substitution or elimination is a true, but worthless, statement (such as 4 4 ), then the system is dependent and has infinite solutions. This occurs when the lines are not distinct – the lines have the same slope and same y-intercept. Substitution and Elimination Methods (Page 11 of 24) The Solution Set to a Linear System of Two Equations in Two Variables has Three Possible Outcomes . . . 1. One Solution See Figure 1. The lines intersect at one point and the solution set contains only one point. A system of linear equations with at least one solution is called a consistent system. The solution to a consistent system is expressed as an ordered pair (a, b) . 2. No Solutions See Figure 2. The lines are distinct and parallel, and never intersect. Therefore, there is no solution. A linear system whose solution set is empty is called an inconsistent system. The Elimination and Substitution methods produce a false statement, such as 4 = 0. y x Figure 1 A Consistent System with one solution. y x Figure 2 An Inconsistent System has no solutions. 3. Infinite Number of Solutions See y Figure 3. The lines lie on top of each other, are not distinct, and have an infinite number of points of intersection – every x point on the line. Therefore, the system of Figure 3 A Dependent equations has an infinite number of System has an infinite solutions. This is called a dependent number of solutions. system. The Elimination and Substitution methods produce a true statement, such as 3 = 3. Using Systems to Model Data (Page 12 of 24) 3.3 Using Systems of Equations to Model Data World Record 400-Meter Run Times Example 1 Women Men Let M (t) and W (t) represent Year Record Time Year Record Time the world record times of (seconds) (seconds) men and women, 1957 57.0 1900 47.8 respectively, for the 4001959 53.4 1916 47.4 meter run at t years since 1962 51.9 1928 47.0 1969 51.7 1932 46.2 1900 (see table). 1972 51.0 1941 46.0 1. Verify the regression 1976 49.29 1950 45.8 equations are 48.60 1960 44.9 W (t) 0.274t 70.454 1979 1983 47.99 1968 43.86 and 1985 47.60 1988 43.29 M (t) 0.053t 48.079 1999 43.18 2. Symbolically (i.e. show your work algebraically) predict when the 400-meter run record time of men and women will be equal. What will that record time be? Verify your result on a graphing calculator using the INTERSECT function. Example 2 In 2002, the price of a 2001 Cadillac De Ville was $30,055, and the price for a 2001 Acura Integra was $16,565. The De Ville depreciates by $4391 per year, and the Integra depreciates by $869 per year. When will the cars have the same value? What will the value be? Show your work (i.e. do not use your calculators). 3.4 Value, Interest, and Mixture Problems (Page 13 of 24) 3.4 VALUE, INTEREST AND MIXTURE PROBLEMS Five-Step Problem Solving Method Step 1 Define each variable. Step 2 Write a system of two equations. Step 3 Solve the system. Step 4 Describe / Explain the result. Step 5 Check your solution. Total Value Formula If n objects have a unit value v, then their total value T is given by T vn i.e. The total value of n nickels is 5n The total value of n dimes is 10n The total value of n quarters is 25n Example 1 A music store charges $5 for a six-string pack of electric-guitar strings and $20 for a four-string pack of electric-bass strings. If the store sells 35 packs of strings for a total of $295, how many packs of each type of string were sold? 3.4 Value, Interest, and Mixture Problems (Page 14 of 24) Example 2 The American Analog Set will play at a auditorium that has 400 balcony seats and 1600 main-level seats. If tickets for balcony seats cost $15 less than the main-level seats, what should the price be for each type of ticket so that the total revenue from a sellout performance will be $70,000? 3.4 Value, Interest, and Mixture Problems (Page 15 of 24) Example 3 A 10,000-seat amphitheater will sell general-seat tickets at $45 and reserved-seat tickets at $65 for Foo Fighters concert. Let x and y be the number of tickets that will sell for $45 and $65, respectively. Assume that the show will sell out. 1. Let T f (x) be the total revenue (in dollars) from selling the tickets. Find the equation for f. 2. Use a graphing calculator to sketch the graph of f for 0 x 10, 000 . What is the slope and what does it mean in this situation? 3. Find f (8500) and explain its meaning in this situation. 4. Find f (11, 000) and explain its meaning in this situation. 5. The total cost of the production is $350,000. How many of each type of ticket must be sold to make a profit of $150,000? 3.4 Value, Interest, and Mixture Problems (Page 16 of 24) Simple Interest Problems If P is the principal, r is the simple annual interest rate, and I is the annual interest earned, then I P r . I Pr Example 4 How much interest will a person earn by investing $3200 in an account at a 4% simple annual interest for one year? Example 5 A person plans to invest twice as much money in an Elfun Trusts account at 2.7% annual interest as in a Vanguard Growth account at 5.5% annual interest. Both interest rates are 5-year averages. How much will the person have to invest in each account to earn a total of $218 in one year? 3.4 Value, Interest, and Mixture Problems (Page 17 of 24) Example 6 A person plans to invest a total of $6000 in Gabelli ABC mutual fund that has a 3-year average annual interest of 6% and in a Presidential Bank Internet CD accou8nt at 2.25% annual interest. Let x and y be the money (in dollars) invested in the mutual fund and CD, respectively. 1. Let I f (x) be the total interest (in dollars) earned from investing the $6000 for one year. Find an equation for f. 2. Use a graphing calculator to draw a graph of f for 0 x 6000 . What is the slope of f and what does it mean in this situation? 3. Use a graphing calculator to create a table of values for f. Explain how such a table can be used to help a person decide how much money to invest in each account. 4. How much money should be invested in each account to earn $300 in one year? 3.4 Value, Interest, and Mixture Problems (Page 18 of 24) Mixture Problems Let c by the percent concentration (in decimal form) of a substance in a solution of volume V, then the quantity Q of the pure substance is given by c V Q e.g. How many ounces of pure lime juice are in 100 oz of a 20% lime solution? Example 7 A chemist needs 5 quarts of a 17% acid solution, but he has only a 15% acid solution and a 25% acid solution. How many quarts of the 15% acid solution should he mix with the 25% acid solution to make 5 quarts of a 17% acid solution? Example 8 A chemist needs 8 cups of a 15% alcohol solution but has only a 20% alcohol solution. How much water and 20% alcohol solution should she mix to form the desired 8 cups of 15% solution? Linear Inequalities in One Variable (Page 19 of 24) 3.5 Linear Inequalities in One Variable A number satisfies an inequality in one variable if substituting the number for the variable results in a true statement. Such numbers are called solutions to the inequality. The solution set to an inequality is the set of all solutions to the inequality. To solve an inequality means to find all the solutions to the inequality. Two inequalities are said to be equivalent if they have the same solutions set. Some example of linear inequalities in one variable follow: 4 x 3 , 3x 2 5 , 8(2 3x) 7 , 3 x5 7 Example 1 1. Is -3 a solution to 8 2x 7 ? 2. Does 6 satisfy the inequality 3(x 2) 6 ? Addition / Subtraction Property of Inequalities Let c be any real number or expression. If a b , then a c b c . In words, adding the same number or expression to both sides of an inequality results in an equivalent inequality. Since subtraction is simply addition of the opposite, c subtracting the same number of c expression from both sides of an a b a+c inequality results in an equivalent inequality (see figure 5). b+c Linear Inequalities in One Variable (Page 20 of 24) Multiplication / Division Property of Inequalities 1. For a positive number c: If a b , then ac bc and e.g. c 3 25 If a b , then ac bc e.g. c 3 5 12 2(3) 5(3) 6 15 5(3) 12(3) 15 36 In words this says multiplying both sides of an inequality by the same positive number results in an equivalent inequality. Similarly, since division is simply multiplication by the reciprocal, dividing both sides of an inequality by the same positive number results in an equivalent inequality. 2. For a negative number c: If a b , then ac bc e.g. c 3 25 2(3) 5(3) 6 15 and If a b , then ac bc e.g. c 3 5 12 5(3) 12(3) 15 36 In words this says multiplying both sides of an inequality the same negative number and reversing the direction of the inequality results in an equivalent inequality. Similarly, since division is simply multiplication by the reciprocal, dividing both sides of an inequality by the same negative number and reversing the direction of the inequality results in an equivalent inequality. Linear Inequalities in One Variable (Page 21 of 24) Example 2 1. Solve 3x 5 7 2. Solve 8 2x 11 3. Solve 3(4 x) 14 x 4. Solve 2[x (2x 3)] 7 2(5 x) 5. Solve 3[7 (2 4x)] 12 7x (3x 8) Linear Inequalities in One Variable (Page 22 of 24) Graphs of Solution Sets and Interval Notation Since the solution set to a linear inequality is generally an infinite set, we can express the solution set as an inequality, a graph, or as an interval using what is referred to as interval notation. In Words Inequality Numbers less than 4 Numbers less than or equal to 4 Numbers greater than 4 Numbers greater than or equal to 4 Numbers between 0 and 4 Numbers between 0 and 4, including 4 Numbers between 0 and 4, including 0 and 4 x4 Graph 0 4 0 4 0 4 0 4 0 4 0x4 0 4 0x4 0 4 x4 x4 x4 0x4 Interval Notation Example 3 Solve 3(4x 5) 1 17 6x . Express the solution set as an inequality, a graph, and in interval notation. Linear Inequalities in One Variable (Page 23 of 24) Example 4 In February 2004, one Budget office rented pickup trucks for $39.95 per day plus $0.19 per mile. One U-Haul location charged $19.95 per day plus $0.49 per mile. For what mileage did Budget offer the lower price? Example 5 Student-to-Teacher The student-to-teacher ratios for Ratios at Public and public and private elementary Private Elementary Schools schools are given in the table. Let Year Public Private f (t) and g(t) represent the student 1995 19.3 16.6 to teacher ratios for public and 1997 18.3 16.6 private elementary schools, 1999 18.1 16.4 2001 17.9 16.3 respectively, at t years since 1990. 1. Verify the regression equations for f and g are f (t) 0.22t 20.16 and g(t) 0.055t 16.915 . 2. Symbolically predict the years in which public schools will have the smaller student-to-teacher ratio. Linear Inequalities in One Variable (Page 24 of 24) Example 6 The graphs of functions f and g are given. Answer the following. a. Find f ( 6) . y b. Find x where g(x) 0 . 5 g -5 5 c. Find the equation for f. f -5 d. Find the equation for g. e. Find x where f (x) g(x) . Express the solution set as an inequality and in interval notation. x