5-11 Systems of Linear and Quadratic Equations

TEKS FOCUS VOCABULARY TEKS (3)(C) Solve, algebraically, systems of two equations in • Formulate – create with careful effort and purpose. two variables consisting of a linear equation and a quadratic You can formulate a plan or strategy to solve a equation. problem.

TEKS (1)(B) Use a problem–solving model that incorporates • Strategy – a plan or method for solving a problem analyzing given information, formulating a plan or strategy, • Reasonableness – the quality of being within the determining a solution, justifying the solution, and evaluating realm of common sense or sound reasoning. The the problem–solving process and the reasonableness of the reasonableness of a solution is whether or not the solution. solution makes sense. Additional TEKS (1)(C), (1)(G), (3)(A), (3)(D)

ESSENTIAL UNDERSTANDING You can solve systems involving quadratic equations using methods similar to the ones used to solve systems of linear equations.

Key Concept Solutions of a Linear-Quadratic System H2_TN H2_TN A system of one quadratic equation and one linear equation can have two solutions, one solution, or no solution.

y = -x2 + 2x + 3 y = -x2 + 2x + 5 y = -x2 + 2x + 5 y 2x 1 y 6 y 1x 9 = + = = -2 + Two solutions One solution No solution

PearsonTEXAS.com 213 Problem 1 Solving a Linear-Quadratic System by Graphing y  x2  5x  6 Multiple Choice Which numbers are y-values of the solutions of the y  x  6 system of equations? { 4 only 6 only 4 and 6 6 and 10 How can you graph these two equations? Graph the equations. Find their intersections. y Use slope-intercept 12 form to graph the linear Note that there are two points where the graphs of equation. Make a table these equations intersect. Then choices A and B are 10 (4, 10) of values to graph the not reasonable solutions. quadratic equation. 8 The solutions appear to be (0, 6) and (4, 10). 6 (0, 6) Check y = -x 2 + 5x + 6 y = x + 6 6 ≟ -(0)2 + 5(0) + 6 6 ≟ 0 + 6 6 = 6 ✔ 6 = 6 ✔ x 2 y = -x + 5x + 6 y = x + 6 Ϫ2 O 2 4 8 10 ≟ -(4)2 + 5(4) + 6 10 ≟ 4 + 6 10 = 10 ✔ 10 = 10 ✔ The y-values of the solutions are 6 and 10, choice D.

Problem 2

Solving a Linear-Quadratic System Using Substitution y  x2  x  6 What is the solution of the system of equations? y  x  3 e

Substitute x 3 for y in + 2 the quadratic equation. x  3  x  x  6

Write in standard form. x 2  2x  3  0

Factor. Solve for x. (x  1)(x  3)  0 x  1 or x  3

Substitute for x in x  1 y  1  3  4 y = x + 3. S x  3 S y  3  3  0 The solutions are (1, 4) and (3, 0).

214 Lesson 5-11 Systems of Linear and Quadratic Equations Problem 3 TEKS Process Standard (1)(C) Solving a Quadratic System of Equations y  x2  x  12 What is the solution of the system? Which variable should y  x2  7x  12 you substitute for? Method 1 Use substitution. e You can substitute for 2 either variable, but Substitute y = -x - x + 12 for y in the second equation. Solve for x. substituting for y results x 2 x 12 x 2 7x 12 Substitute for y. in a simple equation. - - + = + + -2x 2 - 8x = 0 Write in standard form. -2x(x + 4) = 0 Factor. x = 0 or x = -4 Solve for x.

Substitute each value of x into either equation. Solve for y.

y = x 2 + 7x + 12 y = x 2 + 7x + 12 y = (0)2 + 7(0) + 12 y = (-4)2 + 7(-4) + 12 y = 0 + 0 + 12 = 12 y = 16 - 28 + 12 = 0 The solutions are (0, 12) and (-4, 0).

Method 2 Graph the equations.

Use a graphing calculator. Define functions Y1 and Y2.

Plot1 Plot2 Plot3 \Y1 = –X2–X+12 \Y2 = X2+7X+12 \Y3 = \Y4 = \Y5 = \Y6 = \Y7 =

Use the INTERSECT feature to find the points of intersection.

Intersection Intersection X=–4 Y=0 X=0 Y=12

The solutions are (-4, 0) and (0, 12).

PearsonTEXAS.com 215 Problem 4 TEKS Process Standard (1)(B) Formulating a Linear-Quadratic System A Foucault pendulum is used to show the rotation of the Earth. The pendulum swings along a line segment and as the Earth rotates, the endpoints of the segment trace a circle at the base of the pendulum. Though it is not the most precise method of telling time, it can be used to tell the hour of the day with some degree of accuracy. If the center of the circle is at (0, 0) and the radius is 13 units, where does the pendulum intersect the edge of the base when its path (which must 12 pass through the origin) has slope - 5 ?

The base is circular and is modeled The location(s) where Write and solve a system of by a circle with center (0, 0) and the path of the equations representing the radius 13. The swinging of the pendulum intersects the base and path of the pendulum. pendulum at a particular time edge of the base Evaluate the reasonableness of passes through (0, 0) and has the solution(s). 12 slope - 5 .

The equation of a circle with center (0, 0) and radius 13 is x2 + y2 = 169. The line 12 12 passing through (0, 0) with slope - 5 is y = - 5 x. x2 + y2 = 169 Solve the system 12 using substitution. y = - 5 x Why is substitution e a good choice for 12 Step 1 Substitute - 5 x into the first Step 2 Substitute 5 and -5 into the solving the system? equation for y. first equation for x. One of the equations 2 2 2 in the system is already 2 12 x + y = 169 x + ( - 5 x) = 169 solved for y, so you can 2 2 144 ({5) + y = 169 save a step if you use x2 x2 169 + 25 = 2 substitution. y = 144 169x2 169 25 = y = {12 x2 = 25 x = {5 The possible solutions are (5, 12), (5, - 12), ( - 5, 12), and ( - 5, - 12). A line and a circle can intersect in at most two points, so two of the possible solutions are extraneous. The solutions (5, 12) and ( - 5, - 12) are not solutions 12 because they do not solve the equation y = - 5 x. The solutions to the system are (5, - 12) and ( - 5, 12).

216 Lesson 5-11 Systems of Linear and Quadratic Equations NLINE O

H K Scan page for a Virtual Nerd™ tutorial video. O PRACTICE and APPLICATION EXERCISES M R E W O

Solve each system by graphing. Check your answers.

y = -x 2 + 2x + 1 y = x 2 - 2x + 1 y = x 2 - x + 3 1. 2. 3. For additional support when y = 2x + 1 y = 2x + 1 y = -2x + 5 completing your homework, e 2 e 2 e 2 go to PearsonTEXAS.com. y = 2x + 3x + 1 y = -x - 3x + 2 y = -x - 2x - 2 4. 5. 6. y = -2x + 1 y = x + 6 y = x - 4 Solve eeach system by substitution.e Check your answers. e

y = x 2 + 4x + 1 y = -x 2 + 2x + 10 y = -x 2 + x - 1 7. 8. 9. y = x + 1 y = x + 4 y = -x - 1 e e e y = 2x 2 - 3x - 1 y = x 2 - 3x - 20 y = -x 2 - 5x - 1 10. 11. 12. y = x - 3 y = -x - 5 y = x + 2 Solve eeach system. e e

y = x 2 + 5x + 1 y = x 2 - 2x - 1 y = -x 2 - 3x - 2 13. 14. 15. y = x 2 + 2x + 1 y = -x 2 - 2x - 1 y = x 2 + 3x + 2 e e e y = -x 2 - x - 3 y = -3x 2 - x + 2 y = x 2 + 2x + 1 16. 17. 18. y = 2x 2 - 2x - 3 y = x 2 + 2x + 1 y = x 2 + 2x - 1 e e e 19. Apply Mathematics (1)(A) A manufacturer is making cardboard boxes by cutting out four equal squares from the corners of a rectangular piece of cardboard and then folding the remaining part into a box. The length of the cardboard piece is 1 in. longer than its width. The manufacturer can cut out either 3 * 3 in. squares, or 4 * 4 in. squares. Find the dimensions of the cardboard for which the volume of the boxes produced by both methods will be the same. 20. Justify Mathematical Arguments (1)(G) Can you solve the system of equations x = y 2 + 2y + 1 shown by graphing? Justify your answer. Can you solve this system using another y = x - 4 method? If so, solve the system and explain why you chose that method. e Solve each system by substitution. 1 x + y = 3 y - 2x = x + 5 y - x 2 = 1 + 3x 21. 22. 23. 2 y x 2 8x 9 y 1 x2 5x 3 y 1x2 x = - - + = + + + 2 = e e e x + y - 2 = 0 x2 - y = x + 4 2y = y - x2 + 1 24. 25. 26. x2 + y - 8 = 0 x - 1 = y + 3 y = x2 - 5x - 2 e e e 27. Evaluate Reasonableness (1)(B) Your friend wants to solve the system y = x2 and y = x. She concludes that the solution is (0, 0), because 0 = 02. What would you tell your friend about her solution? 28. Create Representations to Communicate Mathematical Ideas (1)(E) A circle with radius of 5 and center at (0, 0) and a line with slope - 1 and y-intercept 7 are graphed on a coordinate plane. What are the solutions to the system of equations?

PearsonTEXAS.com 217 29. Evaluate Reasonableness (1)(B) A system consists of a linear equation and a quadratic equation whose graph is a parabola. Jason says the solutions of the system are (0, 0), (2, 4), and ( - 2, 4). Are Jason’s solutions reasonable? Explain. 30. Use a Problem-Solving Model (1)(B) A water taxi travels around an island in a path that can be modeled by the equation y = 0.5(x - 10)2. A water skier is skiing along a path that begins at the point (6, 5) and ends at the point (8, - 4). a. Write a system of equations to model the problem. b. Is it possible that the water skier could collide with the taxi? Explain. Solve each system. y = 3x 2 - 2x - 1 y = -x 2 + x - 5 y = x 2 - 3x - 2 31. 32. 33. y = x - 1 y = x - 5 y = 4x + 28 e 1 2 e 3 2 e 1 2 y = 2 x + 4x + 4 y = -4 x - 4x y = -4 x + x + 1 34. 1 35. 36. y = -4x + 122 y = 3x + 8 y = x - 4 e e e 37. Apply Mathematics (1)(A) A company’s weekly revenue R is given by the formula R = -p2 + 30p, where p is the price of the company’s product. The company is considering hiring a distributor, which will cost the company 4p + 25 per week. a. Use a system of equations to find the values of the price p for which the product will still remain profitable if they hire this distributor. b. Which value of p will maximize the profit after including the distributor cost? Determine whether the following systems always, sometimes, or never have solutions. (Assume that different letters refer to unequal constants.) Explain. y = x 2 + c y = ax 2 + c y = (x + a)2 y = a(x + m)2 + c 38. 39. 40. 41. y = x 2 + d y = bx 2 + c y = (x + b)2 y = b(x + n)2 + d e e e e

TEXAS Test Practice

1 2 y = -4 x - 2x 42. How many solutions does the system have? 2 3 y = x + 4 A. 0 B. 1 C.e 2 D. 3

43. Which expression is equivalent to (-3 + 2i)(2 - 3i)? F. 13i G. 12 H. 12 + 13i J. -12 44. Which expression is equivalent to (2 - 7i) , (2i)3? 7 1 1 7 7 1 1 7 A. 8 - 4 i B. 4 - 8 i C. 8 + 4 i D. 4 + 8 i 45. Solve the equation -3x 2 + 5x + 4 = 0. Show your work.

218 Lesson 5-11 Systems of Linear and Quadratic Equations