Systems of Linear and Quadratic Equations. Check Skills You ll Need. y x. Solve by Graphing. Solve the following system by graphing.

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1 NY- Learning Standards for Mathematics A.A. Solve a sstem of one linear and one quadratic equation in two variables, where onl factoring is required. A.G.9 Solve sstems of linear and quadratic equations graphicall. Sstems of Linear and Quadratic Equations Check Skills You ll Need Lessons 7-, 7-, and -. Solve the sstem using substitution.. Solve the sstem b graphing. 8. Solve + b factoring. G for Help Solving Sstems Using Graphing In Lesson 7-, ou solved sstems of linear equations graphicall and algebraicall. A sstem of linear equations can have either one solution, no solutions, or infinitel man solutions. In Chapter, ou solved quadratic equations graphicall and algebraicall. In this lesson, ou will stud sstems of linear and quadratic equations. This tpe of sstem can have one solution, two solutions, or no solutions. two solutions one solution no solutions Solve b Graphing Solve the following sstem b graphing. Graph both equations on the same coordinate plane. Identif the point(s) of intersection, if an. The points (, ) and (, ) are the solutions of the sstem. Solve the sstem b graphing. (, ) (, ) NY 7 Chapter NY New York Additional Topics

2 Graphing to Count Solutions Find the number of solutions for the sstem. Step Graph both equations on the same coordinate plane. Step Identif the point(s) of intersection, if an. There are no points of intersection, so there is no solution to the sstem of equations. Find the number of solutions for each sstem. a. b. Solving Sstem Using Algebraic Methods In Lesson 7-, ou solved linear sstems using elimination. The same technique can be applied to sstems of linear and quadratic equations. Using Elimination Solve the following sstem of equations: Step Eliminate. ( ) Subtract the two equations. 7 Subtraction Propert of Equalit Step Factor and solve for 7 ( 9)( 8) Factor. 9 or 8 Zero-Product Propert 9 or 8 Step Find the corresponding values. Use either equation. ( 9) ( 9) (8) (8) The solutions are ( 9, ) and (8, ). Solve the sstem using elimination. See back of book. Lesson NY- Sstems of Linear and Quadratic Equations NY 7

3 Using Substitution Solve the following sstem of equations: 9 and. Step Solve for. Subtract from both sides. Step Write a single equation containing onl one variable. 9 9 Substitute for. ( ) 9 ( ) Subtract from both sides. Step Factor and solve for. ( )( ) Factor. or Zero-Product Propert or Step Find the corresponding -values. Use either equation. ( ) () ( ) () The solutions of the sstem are (, ) and (, ). Solve the sstem using substitution. In Lesson -7, ou used the discriminant to find the number of solutions of a quadratic equation. With sstems of linear and quadratic equations ou can also use the discriminant once ou eliminate a variable. Using the Discriminant to Count Solutions At how man points do the graphs of and 7 intersect? Step Step Eliminate from the sstem. Write the resulting equation in standard form. 7 ( ) Subtract the two equations. Subtraction Propert of Equalit Determine whether the discriminant, b ac, is positive,, or negative. b ac ()() Evaluate the discriminant. Use a, b, and c. Since the discriminant is, there are no solutions. The graphs do not intersect. At how man points do the graphs of and intersect? NY 7 Chapter NY New York Additional Topics

4 Solve Using a Graphing Calculator Solve the sstem of equations and using a graphing calculator. Step Step Step Y=-X+X+ Enter Use the feature. Move the cursor close to and into Select : Intersect. a point of intersection. Y and Y. Press Press three to displa the sstem. First curve? Intersection X= Y= X= Y= times to find the point of intersection. Step Repeat Steps and to find the second intersection point. The solutions of the sstem are (, ) and (, ). Solve the sstem using a graphing calculator. Intersection X= Y= EXERCISES Practice and Problem Solving For more eercises, see Etra Skill and Word Problem Practice. A Eamples and (pages NY 7 and NY 7) G Practice b Eample for Help Eample (page NY 7) Eample (page NY 7) Solve each sstem b graphing. Find the number of solutions for each sstem Solve each sstem using elimination Solve each sstem using substitution Lesson NY- Sstems of Linear and Quadratic Equations NY 7

5 Eample (page NY 7) Eample (page NY 7) Use the discriminant to find the number of solutions for each sstem Solve each sstem using a graphing calculator B Appl Your Skills. Critical Thinking The graph at the right shows a quadratic function and the linear function d. a. If the linear function were changed to d, how man solutions would the sstem have? b. If the linear function were changed to d, how man solutions would the sstem have? Solve each sstem using either elimination or substitution ( ) 8. Graphing Calculator The screen at the right shows the - and -values for the sstem 8 and. Use the table to find the solutions of the sstem. 9. Writing Eplain wh a sstem of linear and quadratic equations cannot have an infinite number of solutions. Use substitution and the quadratic formula to find the solutions of each sstem. Round our answers to the nearest hundredth X X= Y Y The graph at the right shows the sstem and. Find the values of such that the -values on the parabola are units greater than the corresponding -values on the line. Round our answers to the nearest hundredths.. Critical Thinking Solve the sstem and using substitution. How can ou tell that the sstem has no solutions without using graphing, the discriminant, or the quadratic formula? NY 7 Chapter NY New York Additional Topics

6 C Challenge. Geometr The figures below show rectangles that are centered on the -ais with bases on the -ais and upper vertices defined b the function =.. Find the area of each rectangle. Round to the nearest hundredth. a. b. Test Prep c. Find the - and -coordinates of the vertices of the square constructed in the same manner. d. Find the area of the square. Round to the nearest hundredth. REGENTS Multiple Choice Short Response. Which coordinate pair is a solution to the following sstem? A. (, ) B. (, ) C. (, ) D. (, 7) 7. The graph at the right shows the sstem and. How man solutions does the sstem have? F. one solution G. two solutions H. no solutions J. cannot be determined 8. Use the discriminant to determine the number of solutions of the sstem Solve the sstem using substitution and factoring. Show our work. - = Mied Review Lesson NY- Lesson NY- Tell whether each correlation is a causal relationship. Justif our answer.. Hours of computer use and television viewing have a negative correlation. Is this a causation?. The number of ice cream trucks in a town on a given da and the high temperature have a positive correlation. Is this a causation? Find each union or intersection. Let A {,, }, B {,, 8}, C { is an even number less than }, and D { is a multiple of less than }.. A e B. A f B. A f D. C f D. B f C 7. D e C 8. D e A 9. A f C Lesson NY- Sstems of Linear and Quadratic Equations NY 77