Systems of Linear and Quadratic Equations

1 Systems of Linear andQuadratic EquationsLessons 7-1, 7-2, and the system using the system by y 2y 2x 34x y 8x 5x+ 6 0 by Lesson 7-1, you solved Systems of Linear Equations graphically and system of Linear Equations can have either one solution, no solutions, or infinitelymany solutions. In Chapter 10, you solved Quadratic Equations graphically this lesson, you will study Systems of Linear and Quadratic Equations . This type ofsystem can have one solution, two solutions, or no x2 4y x2y x2 4y 3y 0y x 1two solutionsone solutionno solutionsSolve by GraphingSolve the following system by x2 x 2y x 1 Graph both Equations on the same coordinate the point(s) of intersection, if points ( 3, 4) and (1, 0) are the solutions of the the system by 2x 2y x2 x 211 Quick CheckEXAMPLEEXAMPLE11NY-6NY-611 Solving Systems Using GraphingNY 752 Chapter NYNew York Additional TopicsCheck Skills You ll NeedGOGOfor HelpLearning Standards for a system ofone Linear and onequadratic equation intwo variables, where onlyfactoring is required.

2 Systems of linearand Quadratic equationsgraphically. yxO2 2yxO22 24yxO225 2x3O( 3, 4)(1, 0)y 6/13/07 3:50 PM Page 752 Graphing to Count SolutionsFind the number of solutions for the 2x2 3y x 2 Step 1 Graph both Equations on the same coordinate 2 Identify the point(s) of intersection, if are no points of intersection, so there is no solution to the system of the number of solutions for each x x2 6x 10y 2x2 xy 1In Lesson 7-3, you solved Linear Systems using elimination. The same technique canbe applied to Systems of Linear and Quadratic EliminationSolve the following system of Equations :y x2 11x 36y 12x 36 Step x2 11x 36 (y 12x 36)Subtract the two x2 x 72 Subtraction Property of EqualityStep 2 Factor and solve for x0 x2 x 720 (x 9)(x 8) 9 0 or x 8 0 Zero-Product Propertyx 9 or x 8 Step 3 Find the corresponding yvalues. Use either x2 11x 36y x2 11x 36y ( 9)2 11( 9) 36y (8)2 11(8) 36y 81 99 36y 64 88 36y 144y 60 The solutions are ( 9, 144) and (8, 60).

3 Solve the system using x2 4x 1y 3x 133 Quick CheckEXAMPLEEXAMPLE3322 Quick CheckEXAMPLEEXAMPLE22 Lesson NY-6 Systems of Linear and Quadratic EquationsNY 753 See back of System Using Algebraic MethodsOx236 6/13/07 3:50 PM Page 753 Using SubstitutionSolve the following system of Equations :y x2 6x 9 and y x 1 Solvey x 5 for x x 5 xSubtract xfrom both 5 xStep 2 Write a single equation containing only one x2 6x 95 x x2 6x 9 Substitute5 x (5 x) x2 6x 9 (5 x)Subtract 5 xfrom both x2 5x 4 Step 3 Factor and solve for (x 4)(x 1) 4 0orx 1 0 Zero-Product Propertyx 4orx 1 Step 4 Find the corresponding y-values. Use either x2 4x 1y x2 4x 1 (42) 4(4) 1 (12) 4(1) 1 1 4 The solutions of the system are (4, 1) and (1, 4).Solve the system using 30 12xy x2 11x 12In Lesson 10-7, you used the discriminant to find the number of solutions of aquadratic equation. With Systems of Linear and Quadratic Equations you can alsouse the discriminant once you eliminate a the Discriminant to Count SolutionsAt how many points do the graphs of y 2 andy x2 4x 7 intersect?

4 Step 1 Eliminateyfrom the system. Write the resulting equation in standard x2 4x 7 (y 2)Subtract the two x2 4x 5 Subtraction Property of EqualityStep 2 Determine whether the discriminant,b2 4ac, is positive, 0, or 4ac 42 4(1)(5)Evaluate the discriminant. 16 20 Usea 1,b 4, and c 5. 4 Since the discriminant is 4, there are no solutions. The graphs do not how many points do the graphs of y x2 2 and y x 5 intersect?55 Quick CheckEXAMPLEEXAMPLE5544 Quick CheckEXAMPLEEXAMPLE44NY 754 Chapter NYNew York Additional 6/13/07 3:50 PM Page 754 Lesson NY-6 Systems of Linear and Quadratic EquationsNY 755 Solve Using a Graphing CalculatorSolve the system of Equations y x2 4x 1 and y x 5 using a 1 Step 2 Step 3 Entery x2 4x 1 Use the the cursor close to andy x 5 into Select 5: point of and Y2. PressPress three to displaytimes to find the point of the 4 Repeat Steps 2 and 3 to find the second intersection solutions of the system are (1, 4) and (4, 1).

5 Solve the system using a graphing x2 2y xSolve each system by graphing. Find the number of solutions for each x2 x2 x2 5x 4y x 1y 4xy x2 2x x2 2x 3x 4y x 1y 2x 1y x2 Solve each system using x x 7y x2 1y x 2y x2 4x x2 5x x2 x 90y 12xy x2 5x 5y x 30 Solve each system using x2 2x 3x x2 7x 100y 4x 10y x2 34y 10x 3016. x2 x 19 y 3x2 21x 5x y 802x2 y 10x y 1 Example 4(page NY 754)Example 3(page NY 753)Examples 1 and 2(pages NY 752 and NY 753)66 Quick CheckEXAMPLEEXAMPLE66 Practice and Problem SolvingFor more exercises, see Extra Skill and Word Problem by ExampleAAGOGOforHelpY=1Y1=-X2+4X+1 First curve?X=0Y=4 IntersectionX=1Y=1 IntersectionX= 6/13/07 3:50 PM Page 755NY 756 Chapter NYNew York Additional TopicsUse the discriminant to find the number of solutions for each x2 5x x2 3x 6y xy 9 2xy 2x2 25x2 9x x2 4x 20x 29 yy 2 11xy 5x 78x y 20 0 Solve each system using a graphing x2 2x x2 x 5y 2x 2y 4 x2 6x 2x 2x2 24x 7630.

6 X2 8x 15 yy 3 xy 7 11 x y ThinkingThe graph at the right shows a Quadratic function and the Linear function y the Linear function were changed to y d 3, how many solutions would the system have? the Linear function were changed to y d 5, how many solutions would the system have? Solve each system using either elimination or 2x2 x2 9x 91y 9 6xy 1 16x2x 20x x2 12x 14x 1 y15 4x2 9x yy 25(4 x) 12 y 40x CalculatorThe screen at the right shows they- and x-values for the system y x2 6x 8andy x 1. Use the table to find the solutions of the why a system of Linear and Quadratic Equations cannot have an infinite number of substitution and the Quadratic formula to find the solutions of each your answers to the nearest 2x2 4x 4 y 7 5x y 5y x2 4y 6x2 4x graph at the right shows the system y x2 5andy the values ofxsuch that the y-valueson the parabola are 10 units greater than the correspondingy-values on the line. Round your answers to the nearest ThinkingSolve the system y x2 x 25andy xusing substitution.

7 How can you tell that the system has no solutions withoutusing graphing,the discriminant, or the Quadratic formula?y3 Apply Your SkillsBBExample 6(page NY 755)Example 5(page NY 754)yx2424O6 4 2 3X=-1X-101234515830-103-3-2-10123Y1Y24 6128yx 8 6/13/07 10:45 PM Page 756 Lesson NY-6 Systems of Linear and Quadratic EquationsNY figures below show rectangles that are centered on the y-axiswith bases on the x-axis and upper vertices defined by the function y= the area of each rectangle. Round to the nearest the x- and y-coordinates of the vertices of the square constructed in thesame the area of the square. Round to the nearest coordinate pair is a solution to the following system? y x2 2x 2y x 10A.(4, 14)B.(3, 13)C.(2, 12)D.( 3, 7) graph at the right shows the system y x 4 and y x2 many solutionsdoes the system have? be the discriminant to determine the number of solutions of the the system using substitution and factoring.

8 Show your whether each correlation is a causal relationship. Justify your of computer use and television viewing have a negative this a causation? number of ice cream trucks in a town on a given day and the hightemperature have a positive correlation. Is this a causation?Find each union or intersection. Let A {1, 3, 6},B {2, 6, 8},C {x|xis an evennumber less than 6}, and D {x|xis a multiple of 3 less than 12}. NY-4 Lesson NY-5 ChallengeCC Test PrepREGENTSM ultiple ChoiceShort ResponseMixed ReviewMixed Review 262 4 22yx44 262 224 44yxy 49x2 2x 34y 30 100xx2 3x 23 y-5= 6/13/07 3:50 PM Page 757