Solving a Linear-Quadratic System
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1 CC-18 Solving LinearQuadratic Systems Objective Content Standards A.REI.7 Solve a simple system consisting of a linear equation and a quadratic equation in two variables... A.REI.11 Explain why the x-coordinates of the points where the graphs of the equations y f(x) and y g(x) intersect are the solutions of the equation f(x) g(x)... To solve systems of linear and quadratic equations MATHEMATICAL PRACTICES Essential Understanding A linear-quadratic system is made up of a linear equation and a quadratic equation. You can solve this type of system algebraically or graphically. A linear-quadratic system can have zero, one, or two solutions. The line and parabola do not intersect, so the linearquadratic system has no solution. The line and parabola have one point of intersection, so the linear-quadratic system has one solution. The line and parabola have two intersection points, so the linear-quadratic system has two solutions. CC-18 Solving Linear-Quadratic Systems 1
2 Problem 1 Solving a Linear-Quadratic System How is solving this system like solving a system of two linear equations? In both cases, you need to find each point where the graphs of the two equations intersect. What are the solutions of the system? y 2x 2 y x 2 x 6 Method 1: Solve by graphing. Step 1 Graph both equations on the same coordinate plane. Step 2 Identify the point(s) of intersection, if any. The two points of intersection are (, 6) and ( 1, ). The solutions of the system are (, 6) and ( 1, ). 8 y x 8 O 8 8 When solving the system of equations graphically, you can identify the solutions by locating each point of intersection of the two graphs. If you solve the system algebraically, the ordered pair solutions you obtain represent those points of intersection. Method 2: Solve algebraically. Step 1 Write a single equation containing only one variable. y x 2 x 6 2x 2 x 2 x 6 Substitute 2x 2 for y. 2x 2 2x x 2 x 6 2x Subtract 2x from each side. 2 x 2 3x x 2 3x 6 2 Add 2 to each side. Step 2 Factor and solve for x. 0 x 2 3x Write in standard form. 0 (x )(x 1) Factor. x 0 or x 1 0 Zero-Product Property x or x 1 Solve for x. Step 3 Find the corresponding y-values. You may use either of the original equations. y 2x 2 2() 2 6 y 2x 2 2( 1) 2 The solutions of the system are (, 6) and 1, ). Got It? 1. What are the solutions of the system? y x 2 x 3 y x 1 2 Common Core
3 Solutions of systems of equations do not always have integers coordinates. You can estimate or use a graphing calculator tool to find solutions. Problem 2 Estimating Intersection Points What are the solutions of the system? Use a graphing calculator. y x 2 12 x 36 y 3 2 x 5 2 Step 1 Press the y= key and enter the equations. To see the graphs in the standard window, press zoom 6:ZStandard. You may have to press window and adust the window settings to see all of the intersection points for the two graphs. Step 2 Press 2nd trace and choose 5:intersect. Move your cursor close to one of the points of intersection. Press enter three times to find the point of intersection. Intersection X Y How can you check your solutions? You can substitute each x-value into both original equations. Your answers should be close to the estimate for the y-value of the ordered pair. Step 3 Repeat Step 2 to find the second intersection point. The approximate solutions of the system are ( 3.3, 7.) and ( 10.2, 17.8). Using your graphing calculator will only give estimates for the solutions of this system. Algebraic methods for this type of problem can give exact solutions. Intersection X Y Got It? 2. What are the approximate solutions of the system? Use a graphing calculator. y x 2 1 y 3x CC-18 Solving Linear-Quadratic Systems 3
4 Problem 3 Using the Quadratic Formula What are the solutions of the system? Solve by using the quadratic formula. y 3x 7 y 2x 2 7x 10 Step 1 Write a single equation containing only one variable in standard form. 3x 7 2x 2 7x 10 3x 7 3x 2x 2 7x 10 3x Subtract 3x from each side. 7 2x 2 x x 2 x 10 7 Add 7 to each side. 0 2x 2 x 3 Write in standard form. Step 2 Use the quadratic formula to solve. What are the values of a, b, and c? a 2 b c 3 x b b2 ac 2a x ()2 (2)( 3) 2(2) x Step 3 0 Quadratic formula Substitute the values for a, b, and c into the quadratic formula. Simplify. 0 0 x or x Write as two equations. x 0.58 or x 2.58 Use a calculator. Find the corresponding y-values. You may use either of the original equations; however, the linear equation is simpler to use. y 3x 7 y 3(0.58) 7 or y 3( 2.58) 7 Substitute the values for x. y 5.26 or y 1.7 Simplify. The approximate solutions of the system are (0.58, 5.26) and ( 2.58, 1.7). Got It? 3. What are the approximate solutions of the system? y x 2 6x 17 y x 21 Common Core
5 Lesson Check Do you know HOW? 1. Use substitution to solve the system. y x 2 y 2x 3 2. Use a graphing calculator to estimate the solutions of the system. y x 2 2x 3 y 1 2 x 1 3. Use the quadratic formula to solve the system. Round to the nearest hundredth. y x 2 x 1 y 10x 2 Do you UNDERSTAND? MATHEMATICAL PRACTICES. Solve the system y x 2 5x 1 and y 2x 3 using the quadratic formula. What does the value of the discriminant tell you about the number of solutions of the system? 5. Compare and Contrast The solutions of a system of equations solved using the quadratic formula are , 30 2 and , When this system is solved graphically, the solutions are (0.7, 0.12) and (.7, 60.12). Compare the solutions given by each method. Practice and Problem-Solving Exercises MATHEMATICAL PRACTICES A Practice Solve each system of equations. See Problem y x 1 7. y 3x 1 y x 2 x 2 y x 2 x 5 8. y x 1 9. y x 2 8x 5 y x 2 2x 5 y 1x Solve each system using a graphing calculator. Round decimal answers to the nearest hundreth. See Problem y 0.25x y 16x 2 58x 23 y 2x 2 y 12x y 0.5x x 13. y 3x y 0.75x 2 y x x 1 Solve each system using the quadratic formula. Round decimal answers to the nearest hundreth. See Problem y 2x 2 12x y 9x 2 91x 132 y x 1 y 25x y x 17. y x 2 6 y 2x 2 x y 1 CC-18 Solving Linear-Quadratic Systems 5
6 B C Apply Challenge 18. Which method would you use to solve the system y 9 10 x x 20 and y x 11 5? Explain. 19. Reasoning Find a linear equation with a graph that intersects the graph of y x 2 8x 1 at exactly one point. 20. Error Analysis Marge claims that the system y 0.5x x 3 and y 0.5 must have two solutions because all horizontal lines will intersect a parabola in two places. Explain why Marge s thinking is incorrect. 21. Use a graphing calculator to solve x 2 x. 22. The equation for a circle with the center at the origin is given by x 2 y Find where the line y x 1 intersects the circle. 23. Reasoning Given the quadratic function y x 2 1 and the linear function y x b, for what values of b will the system have no solution? Standardized Test Prep SAT/ACT 2. If f (x) (x )(x 6), which equation intersects the graph of f (x) at exactly one point? g(x) g(x) 1 g(x) x g(x) x How many solutions are there for the system of equations y x 2 6x 9 and 2x y 1? Short Response 26. The product of two numbers is 2. If you square the smaller number and add it to the larger number, the sum is 17. Equations modeling the two numbers are shown below. xy 2 x 2 y 17 What is the smaller number? Mixed Review Write an equation in slope-intercept form for each graph shown y y x 8 O 8 8 x 8 O 8 8 Simplify each expression. 29. (x 7 y ) (n) 3 (3n 2 ) 2 6 Common Core
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