Key Concept Solutions of a Linear-Quadratic System

Transcription

1 5-11 Systems of Linear and Quadratic Equations TEKS FOCUS TEKS (3)(C) Solve, algebraically, systems of two equations in two variables consisting of a linear equation and a quadratic equation. TEKS (1)(B) Use a problem solving model that incorporates analyzing given information, formulating a plan or strategy, determining a solution, justifying the solution, and evaluating the problem solving process and the reasonableness of the solution. Additional TEKS (1)(C), (1)(G), (3)(A), (3)(D) VOCABULARY Formulate create with careful effort and purpose. You can formulate a plan or strategy to solve a problem. Strategy a plan or method for solving a problem Reasonableness the quality of being within the realm of common sense or sound reasoning. The reasonableness of a solution is whether or not the solution makes sense. 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 =-x 2 + 2x + 3 y =-x 2 + 2x + 5 y =-x 2 + 2x + 5 y = 2x + 1 y = 6 y =- 1 2 x + 9 Two solutions One solution No solution PearsonTEXAS.com 213

2 Problem 1 How can you graph these two equations? Use slope-intercept form to graph the linear equation. Make a table of values to graph the quadratic equation. Solving a Linear-Quadratic System by Graphing Multiple Choice Which numbers are y-values of the solutions of the system of equations? 4 only 6 only 4 and 6 6 and 10 Graph the equations. Find their intersections. Note that there are two points where the graphs of these equations intersect. Then choices A and B are not reasonable solutions. The solutions appear to be (0, 6) and (4, 10). Check y =-x 2 + 5x + 6 y = x (0) 2 + 5(0) = 6 6 = 6 y =-x 2 + 5x + 6 y = x (4) 2 + 5(4) = = 10 The y-values of the solutions are 6 and 10, choice D y (0, 6) 2 O 2 4 { y = x 2 + 5x + 6 y = x + 6 (4, 10) 8 x Problem 2 Solving a Linear-Quadratic System Using Substitution What is the solution of the system of equations? e y = x2 x + 6 y = x + 3 Substitute x + 3 for y in the quadratic equation. x + 3 = x 2 x + 6 Write in standard form. x 2 + 2x 3 = 0 Factor. Solve for x. Substitute for x in y = x + 3. (x 1)(x + 3) = 0 x = 1 or x = 3 x = 1 S y = = 4 x = 3 S y = = 0 The solutions are (1, 4) and ( 3, 0). 214 Lesson 5-11 Systems of Linear and Quadratic Equations

3 Problem 3 TEKS Process Standard (1)(C) Which variable should you substitute for? You can substitute for either variable, but substituting for y results in a simple equation. Solving a Quadratic System of Equations What is the solution of the system? e y = x2 x + 12 y = x 2 + 7x + 12 Method 1 Use substitution. Substitute y =-x 2 - x + 12 for y in the second equation. Solve for x. -x 2 - x + 12 = x 2 + 7x + 12 Substitute for y. -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 = = 12 y = = 0 The solutions are (0, 12) and (- 4, 0). Method 2 Graph the equations. Use a graphing calculator. Define functions Y 1 and Y 2. Plot1 Plot2 Plot3 \Y1 = X 2 X+12 \Y2 = X 2 +7X+12 \Y3 = \Y4 = \Y5 = \Y6 = \Y7 = Use the INTERSECT feature to find the points of intersection. Intersection X= 4 Intersection Y=0 X=0 Y=12 The solutions are (- 4, 0) and (0, 12). PearsonTEXAS.com 215

4 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 pass through the origin) has slope ? The base is circular and is modeled by a circle with center (0, 0) and radius 13. The swinging of the pendulum at a particular time passes through (0, 0) and has slope The location(s) where the path of the pendulum intersects the edge of the base Write and solve a system of equations representing the base and path of the pendulum. Evaluate the reasonableness of the solution(s). The equation of a circle with center (0, 0) and radius 13 is x 2 + y 2 = 169. The line passing through (0, 0) with slope is y =-12 5 x. Why is substitution a good choice for solving the system? One of the equations in the system is already solved for y, so you can save a step if you use substitution. Solve the system e x2 + y 2 = 169 y = x using substitution. Step 1 Substitute x into the first equation for y. x 2 + ( x ) 2 = 169 x x2 = x2 = 169 x 2 = 25 x = {5 Step 2 Substitute 5 and -5 into the first equation for x. x 2 + y 2 = 169 ( {5) 2 + y 2 = 169 The possible solutions are (5, 12), (5, - 12), ( - 5, 12), and ( - 5, - 12). y 2 = 144 y = {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 because they do not solve the equation y = x. The solutions to the system are (5, - 12) and ( - 5, 12). 216 Lesson 5-11 Systems of Linear and Quadratic Equations

5 ONLINE H O M E W O R K PRACTICE and APPLICATION EXERCISES Scan page for a Virtual Nerd tutorial video. Solve each system by graphing. Check your answers. For additional support when completing your homework, go to PearsonTEXAS.com. 1. e y =-x 2 + 2x + 1 y = 2x e y = 2x 2 + 3x + 1 y =-2x e y = x 2-2x + 1 y = 2x e y =-x 2-3x + 2 y = x + 6 Solve each system by substitution. Check your answers. 3. e y = x 2 - x + 3 y =-2x e y =-x 2-2x - 2 y = x e y = x 2 + 4x + 1 y = x e y = 2x 2-3x - 1 y = x - 3 Solve each system. 13. e y = x 2 + 5x + 1 y = x 2 + 2x e y =-x 2 - x - 3 y = 2x 2-2x e y =-x 2 + 2x + 10 y = x e y = x 2-3x - 20 y =-x e y = x 2-2x - 1 y =-x 2-2x e y =-3x 2 - x + 2 y = x 2 + 2x e y =-x 2 + x - 1 y =-x e y =-x 2-5x - 1 y = x e y =-x 2-3x - 2 y = x 2 + 3x e y = x 2 + 2x + 1 y = x 2 + 2x 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 shown by graphing? Justify your answer. Can you solve this system using another method? If so, solve the system and explain why you chose that method. Solve each system by substitution. e x = y 2 + 2y + 1 y = x e x + y = 3 y = x 2-8x e x + y - 2 = 0 x 2 + y - 8 = e y - 2x = x + 5 y + 1 = x 2 + 5x e x2 - y = x + 4 x - 1 = y e y x 2 = 1 + 3x y x2 = x 26. e 2y = y - x2 + 1 y = x 2-5x Evaluate Reasonableness (1)(B) Your friend wants to solve the system y = x 2 and y = x. She concludes that the solution is (0, 0), because 0 = 0 2. 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

6 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. 31. e y = 3x 2-2x e y =-x 2 + x e y = x 2-3x - 2 y = x - 1 y = x - 5 y = 4x e y = 1 2 x 2 + 4x + 4 y =-4x e y =-3 4 x 2-4x 36. e y =-1 4 x 2 + x y = 3x + 8 y = x Apply Mathematics (1)(A) A company s weekly revenue R is given by the formula R =-p p, 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. 38. e y = x 2 + c y = x 2 + d 39. e y = ax 2 + c y = bx 2 + c y = (x + a)2 40. e y = (x + b) ey = a(x + m)2 + c y = b(x + n) 2 + d TEXAS Test Practice 42. How many solutions does the system have? e y =-1 4 x 2-2x y = x A. 0 B. 1 C. 2 D Which expression is equivalent to ( i)(2-3i)? F. 13i G. 12 H i J Which expression is equivalent to (2-7i), (2i) 3? A i B i C i D i 45. Solve the equation -3x 2 + 5x + 4 = 0. Show your work. 218 Lesson 5-11 Systems of Linear and Quadratic Equations