Solving Linear-Quadratic Systems
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1 36 LESSON Solving Linear-Quadratic Sstems UNDERSTAND A sstem of two or more equations can include linear and nonlinear equations. In a linear-quadratic sstem, there is one linear equation and one quadratic equation, each in two variables, usuall and. Recall that the solution to a sstem of linear equations is the ordered pair that satisfies ever equation in the sstem. The same is true for the solution or solutions to a linear-quadratic sstem. An (, ) pair that satisfies both equations in the sstem is a solution to that sstem. A linear-quadratic sstem can have zero, one, or two real solutions. An point where the graphs of the equations intersect is a solution to the sstem. If the line representing the linear equation crosses the graph of the quadratic equation in two places, the sstem has two real solutions. If the line is tangent to the graph of the quadratic equation, the sstem has onl one solution. If the line does not intersect the graph of the quadratic equation at all, the sstem has no real solution solutions 1 solution no solution Though onl parabolas are shown, the statements above appl to a sstem consisting of a linear equation and an quadratic equation, including equations whose graphs are circles. To solve a linear-quadratic sstem using algebra, use the substitution method. Rewrite the linear equation to isolate one of the variables. Then substitute the equivalent epression for that variable into the quadratic equation. The result will be a quadratic equation in one variable. From that point, ou can use an of the methods ou have learned to solve for the value(s) of that variable. These include factoring, taking the square root of both sides, completing the square, and using the quadratic formula. Once ou know the value or values of that variable, ou can substitute each of them back into one of the original equations in order to find the corresponding value of the other variable. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 86 Unit : Epressions and Equations
2 Connect Solve the sstem using algebra: Solve the linear equation for Use substitution to produce an equation in one variable. Since 3, substitute this epression for in the quadratic equation. 3 Solve for the remaining variable. Rewrite the equation in standard form The result is a quadratic equation with onl one variable,. Duplicating this page is prohibited b law. 01 Triumph Learning, LLC CHECK Factor the equation and use the zero product propert ( )( 11) 0 or Use our graphing calculator to graph the equations. Then, check the solution b using the TABLE feature or the intersect function in the CALCULATE menu. Find the values of the other variable. Substitute the values of into the linear equation to find the corresponding values of. For : For 11: 3( ) 3( 11) The solutions to the sstem are (, 16) and ( 11, 37). Lesson 36: Solving Linear-Quadratic Sstems 87
3 EXAMPLE Use algebra to find the points of intersection of the line represented b the equation 6 and the circle represented b the equation ( 3) ( 1). 1 Substitute to form a quadratic equation in one variable. Put the quadratic equation in standard form. The points of intersection of the two graphs are the solutions to the sstem containing both equations. The linear equation is alread solved for, so substitute the equivalent epression, 6, for into the quadratic equation. Epand both of the squared binomials. Then combine them. ( 3) ( 5) ( 6 9) (16 0 5) ( 16 ) ( 6 0) (9 5) ( 3) ( 1) 3 3 ( 3) ( 6 1) ( 3) ( 5) Subtract from both sides to put the equation in standard form Find the solutions. The terms have a GCF of, so divide both sides of the equation b TRY Solve the sstem below b graphing both equations ( ) ( ) 9 This is a perfect square trinomial. Write it as a squared binomial and solve for. ( 1) 0 ( 1) Substitute 1 for to find. 6 ( 1) 6 The sstem has one solution, ( 1, ). Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 88 Unit : Epressions and Equations
4 READ Problem Solving A model rocket is launched straight up with an initial velocit of 80 ft/s. At the same time, a bird is fling overhead at an altitude of 89.6 ft. The bird is descending 1.6 feet ever second as it flies. At what times will the rocket and the bird have the same elevation? PLAN Write one equation for the height, h, of the bird at time t. Write another equation for the height, h, of the rocket at time t. Solve the sstem containing these two equations. SOLVE The bird s change in height is 1.6 ft/s, and its height at time t 0 is 89.6 ft. So, the equation for the bird s elevation is t The model rocket has an initial velocit of 80 ft/s and an initial height of 0 ft since it is on the ground. So, the equation for the rocket s elevation is h 16t t. Substitute for h. Then put the equation in standard form. t t t 0 16t t 89.6 Use the formula to solve for t. t ( ) (81.6) ( 16)( ) ( ) CHECK t or t Duplicating this page is prohibited b law. 01 Triumph Learning, LLC Substitute each value of t into both equations to confirm that it ields the same value of h. For t : ( ) 89.6 h 16( ) ( ) For t : ( ) 89.6 h 16( ) ( ) The bird and rocket will the same height after seconds and seconds. Lesson 36: Solving Linear-Quadratic Sstems 89
5 Practice Solve the sstems of equations b graphing. Give our answers as ordered pairs. 1. ( ) ( 1) REMEMBER Find the points of intersection. Choose the best answer. 3. Which is/are the solution(s) to the following sstem? 6 8. Gina correctl graphed this sstem on her graphing calculator: 6 3 A. (, 0) B. (, 0), (6, 8) C. (8, 6) D. (0, ), (8, 6) HINT Substitute the coordinate pairs into both equations. A picture of her calculator screen is shown below. How man solutions does this sstem have? A. 0 C. B. 1 D. 3 Duplicating this page is prohibited b law. 01 Triumph Learning, LLC 90 Unit : Epressions and Equations
6 Solve the sstems of equations algebraicall. Give our answers as ordered pairs Solve. 8. The height of a ball t seconds after it was thrown is modeled b the equation h 16t 67t, where h is its height in feet above the ground. At the same time, a bird fling through the air had a height of h 3t 8. Solve the sstem of two equations to find the time(s) when the ball and the bird were at the same elevation, and find that elevation. 9. DRAW A fenced-in, square park is 30 feet on each side. The entrance to the park is in the center of the south edge of the park. At the center of the park is a fountain with a diameter of 10 feet. Luis is meeting Samantha in the park. Samantha is 5 feet north and 5 feet west of the entrance. Can Luis walk a straightline path from the entrance to where Samantha is sitting? Eplain our answer INTERPRET The Jump Shot compan sells basketballs. The amount of mone, M, that the compan takes in from selling b basketballs per da is modeled b the equation M b 0b. The amount of mone, M, that it costs the compan to make b basketballs per da is modeled b the equation M 10b 100. Solve the sstem: M b 0b M 10b 100 Duplicating this page is prohibited b law. 01 Triumph Learning, LLC If profit is equal to the amount of mone made minus the cost, does the compan make a profit at these solution point(s)? Wh or wh not? If the compan sells 7 basketballs per da, is it making a profit? How do ou know? Lesson 36: Solving Linear-Quadratic Sstems 91
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