3-1. Solving Systems Using Tables and Graphs. Concept Summary. Graphical Solutions of Linear Systems VOCABULARY TEKS FOCUS ESSENTIAL UNDERSTANDING

Transcription

1 3- Solving Sstems Using Tables and Graphs TEKS FOCUS VOCABULARY Foundational to TEKS (3)(A) Formulate sstems of equations, including sstems consisting of three linear equations in three variables and sstems consisting of two equations, the first linear and the second quadratic. TEKS ()(D) Communicate mathematical ideas, reasoning, and their implications using multiple representations, including smbols, diagrams, graphs, and language as appropriate. Consistent sstem A sstem of linear equations is consistent if it has at least one solution. Dependent sstem A sstem of equations that has an infinite set of solutions is a dependent sstem. Inconsistent sstem A sstem of equations that has no solution is an inconsistent sstem. Independent sstem A sstem Additional TEKS ()(A), (8)(B) of linear equations that has a unique solution is an independent sstem. Linear sstem A linear sstem is a set of two or more linear equations that use the same variables. Solution of a sstem A solution of a sstem is a set of values for the variables that makes all the equations true. Sstem of equations A sstem of equations is a set of two or more equations using the same variables. Implication a conclusion that follows from previousl stated ideas or reasoning without being eplicitl stated Representation a wa to displa or describe information. You can use a representation to present mathematical ideas and data. ESSENTIAL UNDERSTANDING To solve a sstem of equations, find a set of values that replace the variables in the equations and make each equation true. Concept Summar Intersecting Lines Graphical Solutions of Linear Sstems Coinciding Lines O O one solution consistent independent 70 Parallel Lines infinitel man solutions consistent dependent Lesson 3- Solving Sstems Using Tables and Graphs O no solution inconsistent

2 Problem TEKS Process Standard ()(D) How can ou use a graph to find the solution of a sstem? Find the point where the two lines intersect. Using a Graph or Table to Solve a Sstem 3 8 What is the solution of the sstem? b 8 Method Graph the equations. The point of intersection appears to be (-4, -). Check b substituting the values into both equations = 8 + = -8-3(-4) + (-) = (-) = -8 Both equations are true, so (-4, -) is the solution of the sstem. Method Use a table. Write the equations in slope-intercept form = 8 + = -8 = = = = Enter the equations in the Y screen as Y and Y. View the table. Adjust the -values until ou see =. When = -4, both and equal -. So, (-4, -) is the solution of the sstem O ( 4, ) 8 X Y Y X = -4 Problem Using a Table to Solve a Problem STEM Biolog The diagrams show the birth lengths and growth rates of two species of shark. If the growth rates sta the same, at what age would a Spin Dogfish and a Greenland shark be the same length? GREENLAND SHARK How can ou use slope-intercept form to write each equation? Use the growth rate for m and the length at birth for b. Step Define the variables and write the equation for the length of each shark. Let = age in ears. Let = length in centimeters. Length of Greenland: = Length of Spin Dogfish: =.5 + Growth rate: 0.75 cm/r Birth length: 37 cm SPINY DOGFISH SHARK Growth rate:.5 cm/r Birth length: cm continued on net page PearsonTEXAS.com 7

3 Problem continued Step Use the table to solve the problem. List -values until the corresponding -values match. The sharks will be the same length when the are 0 ears old. Age Shark Length in cm Greenland Spin Dogfish Problem 3 Using Linear Regression TEKS Process Standard ()(A) Population The table shows the populations of the New York Cit and Los Angeles metropolitan regions from the census reports for 950 through 000. Populations of New York Cit and Los Angeles Metropolitan Regions ( ) New York Cit Los Angeles 950,9,994 4,367, ,759,49 6,74, ,78,700 7,03, ,,97,497, ,087,5 4,53,59 000,99,865 6,373,645 SOURCE: U.S. Census Bureau Assuming these linear trends continue, when will the populations of these regions be equal? What will that population be? Population data for two regions The point in time when their populations will be the same Use a calculator to find linear regression models. Plot the models. Find the point of intersection. Enter all the numbers as millions, rounded to the nearest hundred thousand. For eample, enter,9,994 as.9. Step Step Enter the data into lists on our calculator. L: number of ears since 950 L: New York Cit populations L3: Los Angeles populations Use LinReg(a b) to find lines of best fit. Use L and L for New York Cit. Use L and L3 for Los Angeles. L L()=0 L L continued on net page 7 Lesson 3- Solving Sstems Using Tables and Graphs

4 Problem 3 continued What does represent? The -value is the number of ears since the zero ear. Step 3 Graph the linear regression lines. Use the Intersect feature. The -value of the point of intersection is about 87, which represents the ear 037. The data suggest that the populations of the New York Cit and Los Angeles metropolitan regions will each be about 5.6 million in 037. Problem 4 Classifing a Sstem Without Graphing What should ou compare to classif the sstem? Compare the slopes and -intercepts of each line. Without graphing, is the sstem independent, dependent, or inconsistent? 4 = 6 b 8 = 4 Rewrite each equation in slope-intercept form. Compare slopes and -intercepts. 4 - = 6 8 = 4 - = + 3 = - 3 m = ; -intercept is 3 m = ; -intercept is - 3 The slopes are equal and the -intercepts are different. The lines are different but parallel. The sstem is inconsistent. ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. For additional support when completing our homework, go to PearsonTEXAS.com. Use Multiple Representations to Communicate Mathematical Ideas ()(D) Solve each sstem b graphing or using a table. Check our answers.. b = - = = 3. b + =. b = = 3-4. b = -3 = 5 - = 4 5. b - = 6 6. b 3 + = 5 - = 7 7. You and a friend are both reading a book. You read pages each minute and have alread read 55 pages. Your friend reads 3 pages each minute and has alread read 35 pages. Graph and solve a sstem of equations to find when the two of ou will have read the same number of pages. Since the number of pages ou have read depends on how long ou have been reading, let represent the number of minutes it takes to read pages. PearsonTEXAS.com 73

5 Write and solve a sstem of equations for each situation. Check our answers. 8. A store sells small notebooks for $8 and large notebooks for $0. If ou bu 6 notebooks and spend $56, how man of each size notebook did ou bu? 9. A shop has bags of peanuts for sale. If ou bu 5 bags and spend $7, how man of each size bag did ou bu? 0. Appl Mathematics ()(A) You can choose between two tennis courts at two universit campuses to learn how to pla tennis. One campus charges $5 per hour. The other campus charges $0 per hour plus a one-time registration fee of $0. a. Write a sstem of equations to represent the cost c for h hours of court use at each campus. b. Find the number of hours for which the costs are the same. c. If ou want to practice for a total of 0 hours, which universit campus should ou choose? Eplain. Without graphing, classif each sstem as independent, dependent, or inconsistent. 74. b 7 - = = -6. b -3 + = 4-3 = 3. b = + = b = - = b =6 = - 6. b = = 8 Lesson 3- Solving Sstems Using Tables and Graphs

6 Select Tools to Solve Problems ()(C) Find linear models for each set of data. In what ear will the two quantities be equal? 7. U.S. Life Epectanc at Birth ( ) Year Men (ears) Women (ears) SOURCE: U.S. Census Bureau 8. Annual U.S. Consumption of Vegetables Year Broccoli (lb/person) Cucumbers (lb/person) SOURCE: U.S. Census Bureau 9. Use Representations to Communicate Mathematical Ideas ()(E) Your friend used a graphing calculator to solve a sstem of linear equations, shown below. After using the TABLE feature, our friend sas that the sstem has no solution. Eplain what our friend did wrong. What is the solution of the sstem? + = 6 = = = X Y Y X= 0. Eplain Mathematical Ideas ()(G) Is it possible for an inconsistent linear sstem to contain two lines with the same -intercept? Eplain.. Connect Mathematical Ideas ()(F) Summarize the possible relationships for the -intercepts, slopes, and number of solutions in a sstem of two linear equations in two variables. Write a second equation for each sstem so that the sstem will have the indicated number of solutions.. infinite number of solutions 3. no solutions b = 5 + = 0 b?? 4. Write a sstem of linear equations with the solution set 5(, ) 0 = Analze Mathematical Relationships ()(F) What relationship eists between the equations in a dependent sstem? PearsonTEXAS.com 75

7 6. Appl Mathematics ()(A) Research shows that in a certain market onl 000 widgets can be sold at $8 each, but if the price is reduced to $3, then 0,000 can be sold. a. Let p represent price and n represent the number of widgets. Identif the independent and dependent variables. b. Write a linear equation that relates price and the quantit demanded. This tpe of equation is called a demand equation. c. A shop can make 000 widgets for $5 each and 0,000 widgets for $ each. Use this information to write a linear equation that relates price and the quantit supplied. This tpe of equation is called a suppl equation. d. Find the equilibrium point where suppl is equal to demand. Eplain the meaning of the coordinates of this point within the contet of the eercise. TEXAS Test Practice 7. Which graph shows the solution of the following sstem? b 4 + = + 4 = - A. C. O O B. D. O O 8. Which is the equation of a line that is perpendicular to the line in the graph? F. = -3 + H. = G. = J. = 3 - O 9. Am ordered prints of a total of 6 photographs in two different sizes, 5 * 7 and 4 * 6, from an online site. She paid $7.50 for her order. The cost of a 5 * 7 print is $.75 and the cost of a 4 * 6 print is $.5. Eplain how to solve a sstem of equations using tables to find the number of 4 * 6 prints Am ordered. 76 Lesson 3- Solving Sstems Using Tables and Graphs