Solving Special Systems of Linear Equations

Transcription

1 5. Solving Special Sstems of Linear Equations Essential Question Can a sstem of linear equations have no solution or infinitel man solutions? Using a Table to Solve a Sstem Work with a partner. You invest $5 for equipment to make skateboards. The materials for each skateboard cost $. You sell each skateboard for $. a. Write the cost and revenue equations. Then cop and complete the table for our cost C and our revenue R. (skateboards) C (dollars) R (dollars) MODELING WITH MATHEMATICS To be proficient in math, ou need to interpret mathematical results in real-life contets. b. When will our compan break even? What is wrong? Writing and Analzing a Sstem Work with a partner. A necklace and matching bracelet have two tpes of beads. The necklace has small beads and large beads and weighs grams. The bracelet has small beads and 3 large beads and weighs 5 grams. The threads holding the beads have no significant weight. a. Write a sstem of linear equations that represents the situation. Let be the weight (in grams) of a small bead and let be the weight (in grams) of a large bead. b. Graph the sstem in the coordinate plane shown. What do ou notice about the two lines? c. Can ou find the weight of each tpe of bead? Eplain our reasoning..5 Communicate Your Answer 3. Can a sstem of linear equations have no solution or infinitel man solutions? Give eamples to support our answers.. Does the sstem of linear equations represented b each graph have no solution, one solution, or infi nitel man solutions? Eplain a. = + b. = + c. = + + = 3 + = 3 + = Section 5. Solving Special Sstems of Linear Equations 53

2 5. Lesson What You Will Learn Core Vocabular Previous parallel Determine the numbers of solutions of linear sstems. Use linear sstems to solve real-life problems. The Numbers of Solutions of Linear Sstems Core Concept Solutions of Sstems of Linear Equations A sstem of linear equations can have one solution, no solution, or infi nitel man solutions. One solution No solution Infinitel man solutions ANOTHER WAY You can solve some linear sstems b inspection. In Eample, notice ou can rewrite the sstem as + = + = 5. This sstem has no solution because + cannot be equal to both and 5. The lines intersect. The lines are parallel. The lines are the same. Solving a Sstem: No Solution Solve the sstem of linear equations. = + Equation = 5 Equation SOLUTION Method Solve b graphing. Graph each equation. The lines have the same slope and different -intercepts. So, the lines are parallel. = + Because parallel lines do not intersect, there is no point that is a solution of both equations. = 5 So, the sstem of linear equations has no solution. Method Solve b substitution. Substitute 5 for in Equation. STUDY TIP A linear sstem with no solution is called an inconsistent sstem. = + Equation 5 = + Substitute 5 for. 5 = Subtract from each side. The equation 5 = is never true. So, the sstem of linear equations has no solution. 5 Chapter 5 Solving Sstems of Linear Equations

3 Solving a Sstem: Infinitel Man Solutions Solve the sstem of linear equations. + = 3 Equation + = Equation SOLUTION Method Solve b graphing. Graph each equation. + = + = 3 The lines have the same slope and the same -intercept. So, the lines are the same. Because the lines are the same, all points on the line are solutions of both equations. So, the sstem of linear equations has infinitel man solutions. Method Solve b elimination. Step Multipl Equation b. + = 3 Multipl b. = Revised Equation + = + = Equation STUDY TIP A linear sstem with infinitel man solutions is called a consistent dependent sstem. Step Add the equations. = Revised Equation + = Equation = Add the equations. The equation = is alwas true. So, the solutions are all the points on the line + = 3. The sstem of linear equations has infinitel man solutions. Monitoring Progress Help in English and Spanish at BigIdeasMath.com Solve the sstem of linear equations.. + = 3. = = + = 3. + = 3. = + + = + = Section 5. Solving Special Sstems of Linear Equations 55

4 Solving Real-Life Problems Modeling with Mathematics The perimeter of the trapezoidal piece of land is 8 kilometers. The perimeter of the rectangular piece of land is kilometers. Write and solve a sstem of linear equations to find the values of and. 8 SOLUTION. Understand the Problem You know the perimeter of each piece of land and the side lengths in terms of or. You are asked to write and solve a sstem of linear equations to find the values of and Make a Plan Use the figures and the definition of perimeter to write a sstem of linear equations that represents the problem. Then solve the sstem of linear equations Solve the Problem Perimeter of trapezoid Perimeter of rectangle = = + = 8 Equation = Equation Sstem + = 8 Equation = Equation Method Solve b graphing. Graph each equation. The lines have the same slope and the same -intercept. So, the lines are the same. In this contet, and must be positive. Because the lines are the same, all the points on the line in Quadrant I are solutions of both equations = + = 8 So, the sstem of linear equations has infinitel man solutions. Method Solve b elimination. Multipl Equation b and add the equations. + = 8 Multipl b. 8 3 = Revised Equation = = Equation = Add the equations. The equation = is alwas true. In this contet, and must be positive. So, the solutions are all the points on the line + = 8 in Quadrant I. The sstem of linear equations has infinitel man solutions.. Look Back Choose a few of the ordered pairs (, ) that are solutions of Equation. You should find that no matter which ordered pairs ou choose, the will also be solutions of Equation. So, infi nitel man solutions seems reasonable. Monitoring Progress Help in English and Spanish at BigIdeasMath.com 5. WHAT IF? What happens to the solution in Eample 3 when the perimeter of the trapezoidal piece of land is 9 kilometers? Eplain. 5 Chapter 5 Solving Sstems of Linear Equations

5 5. Eercises Dnamic Solutions available at BigIdeasMath.com Vocabular and Core Concept Check. REASONING Is it possible for a sstem of linear equations to have eactl two solutions? Eplain.. WRITING Compare the graph of a sstem of linear equations that has infinitel man solutions and the graph of a sstem of linear equations that has no solution. Monitoring Progress and Modeling with Mathematics In Eercises 3 8, match the sstem of linear equations with its graph. Then determine whether the sstem has one solution, no solution, or infinitel man solutions. A =. = = + = 5. + =. = = 8 5 = 7. + = = 7 3 = 9 3 = B = = = + = =. 3 = 5 = + 5 = 7 In Eercises 7, use onl the slopes and -intercepts of the graphs of the equations to determine whether the sstem of linear equations has one solution, no solution, or infinitel man solutions. Eplain. 7. = = + 3 = 39 + = = = 3 = 7 = =. = 3 = 3 = 3 C. D. ERROR ANALYSIS In Eercises 3 and, describe and correct the error in solving the sstem of linear equations. E. F = + = In Eercises 9, solve the sstem of linear equations. (See Eamples and.) 9. =. = 8 = = =. + = 7 + = = 7. The lines do not intersect. So, the sstem has no solution. = 3 8 = 3 The lines have the same slope. So, the sstem has infinitel man solutions. Section 5. Solving Special Sstems of Linear Equations 57

6 5. MODELING WITH MATHEMATICS A small bag of trail mi contains 3 cups of dried fruit and cups of almonds. A large bag contains cups of dried fruit and cups of almonds. Write and solve a sstem of linear equations to find the price of cup of dried fruit and cup of almonds. (See Eample 3.) $9 $. MODELING WITH MATHEMATICS In a canoe race, Team A is traveling miles per hour and is miles ahead of Team B. Team B is also traveling miles per hour. The teams continue traveling at their current rates for the remainder of the race. Write a sstem of linear equations that represents this situation. Will Team B catch up to Team A? Eplain. 7. PROBLEM SOLVING A train travels from New York Cit to Washington, D.C., and then back to New York Cit. The table shows the number of tickets purchased for each leg of the trip. The cost per ticket is the same for each leg of the trip. Is there enough information to determine the cost of one coach ticket? Eplain. Destination Coach tickets Business class tickets Mone collected (dollars) Washington, D.C. 5 8,8 New York Cit 7 7,8 8. THOUGHT PROVOKING Write a sstem of three linear equations in two variables so that an two of the equations have eactl one solution, but the entire sstem of equations has no solution. 3. HOW DO YOU SEE IT? The graph shows information about the last leg of a -meter rela for three rela teams. Team A s runner ran about 7.8 meters per second, Team B s runner ran about 7.8 meters per second, and Team C s runner ran about 8.8 meters per second. Distance (meters) 5 Last Leg of -Meter Rela 5 Team A Team C Team B 8 8 Time (seconds) a. Estimate the distance at which Team C s runner passed Team B s runner. b. If the race was longer, could Team C s runner have passed Team A s runner? Eplain. c. If the race was longer, could Team B s runner have passed Team A s runner? Eplain. 3. ABSTRACT REASONING Consider the sstem of linear equations = a + and = b, where a and b are real numbers. Determine whether each statement is alwas, sometimes, or never true. Eplain our reasoning. a. The sstem has infinitel man solutions. b. The sstem has no solution. c. When a < b, the sstem has one solution. 3. MAKING AN ARGUMENT One admission to an ice skating rink costs dollars, and renting a pair of ice skates costs dollars. Your friend sas she can determine the eact cost of one admission and one skate rental. Is our friend correct? Eplain. 9. REASONING In a sstem of linear equations, one equation has a slope of and the other equation has a slope of 3. How man solutions does the sstem have? Eplain. Maintaining Mathematical Proficienc 3 Admissions Skate Rentals Solve the equation. Check our solutions. (Section.) = = = = 3 5 Admissions Skate Rentals Total $ 38. Total $ 9. Reviewing what ou learned in previous grades and lessons 58 Chapter 5 Solving Sstems of Linear Equations