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Introduction

Introduction A system of equations is a set of equations with the same unknowns. It is likely that you have solved a system of equations or a quadratic-linear system in the past. In this lesson, we will learn to solve systems that include at least one polynomial by graphing. Key Concepts

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Introduction

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  1. Introduction A system of equations is a set of equations with the same unknowns. It is likely that you have solved a system of equations or a quadratic-linear system in the past. In this lesson, we will learn to solve systems that include at least one polynomial by graphing. 2.4.1: Solving a Linear-Polynomial System of Equations

  2. Key Concepts The real solutions of a system of equations can often be found by graphing. Recall that the graph of an equation is the set of all the solutions of the equation plotted on a coordinate plane. The points of intersection of the graphed system of equations are the ordered pairs where graphed functions intersect on a coordinate plane. These are also the solutions to the system. The solution or solutions to a system of equations with f(x) and g(x) occurs where f(x) = g(x) and the difference between these two values is 0. Also recall that the solutions to a system of equations are written using braces: {}. 2.4.1: Solving a Linear-Polynomial System of Equations

  3. Key Concepts, continued A system may have an infinite number of solutions, a finite number of solutions, or no real solutions. A system with a finite number of points of intersection is called an independent system. It is possible that a system of equations does not intersect. The solution to such a system of equations is the empty set, denoted by { }. A system of equations could also intersect at every point. This type of system is known as a dependent system. 2.4.1: Solving a Linear-Polynomial System of Equations

  4. Key Concepts, continued Solutions to Systems of Equations with Polynomials • If the equations overlap, they have an infinite number of solutions. • This is an example of a dependent system. 2.4.1: Solving a Linear-Polynomial System of Equations

  5. Key Concepts, continued Solutions to Systems of Equations with Polynomials • If the equations intersect, they have a finite number of solutions. • This is an example of an independent system. 2.4.1: Solving a Linear-Polynomial System of Equations

  6. Key Concepts, continued Solutions to Systems of Equations with Polynomials • If the equations do not intersect, they have no solutions. • The solution to this system is the empty set. 2.4.1: Solving a Linear-Polynomial System of Equations

  7. Key Concepts, continued When graphing a system of equations with polynomials, it is helpful to recall various forms of equations and their general shapes. 2.4.1: Solving a Linear-Polynomial System of Equations

  8. Key Concepts, continued Graphs of General Equations 2.4.1: Solving a Linear-Polynomial System of Equations

  9. Key Concepts, continued 2.4.1: Solving a Linear-Polynomial System of Equations

  10. Key Concepts, continued 2.4.1: Solving a Linear-Polynomial System of Equations

  11. Key Concepts, continued To graph a system of equations, create a table of values and plot each coordinate on the same coordinate plane. It may be necessary to find additional points to determine where the point(s) of intersection may occur. Each point of intersection represents a solution to the system. Solutions to a system of equations can also be identified in a table by finding points such that the y-value of each function is the same for a particular x-value. 2.4.1: Solving a Linear-Polynomial System of Equations

  12. Key Concepts, continued Using a graphing calculator to graph both equations is often helpful. Most calculators have a trace feature to approximate where the equations intersect, or can display a table of values for each equation so you can analyze the values. Some calculators will find intersection points for you. To confirm a solution to a system, substitute the coordinates for x and y in both of the original equations. If the intersection point is approximated, the results will be nearly equal. 2.4.1: Solving a Linear-Polynomial System of Equations

  13. Key Concepts, continued Solving Systems of Equations Algebraically When solving a quadratic-linear system, if both functions are written in the form of a function (such as “y =” or “f(x) =”), set the equations equal to each other. When you set the equations equal to each other, you are replacing y in each equation with an equivalent expression, thus using the substitution method to find a solution. 2.4.1: Solving a Linear-Polynomial System of Equations

  14. Key Concepts, continued Recall that you can then solve by factoring the equation or by using the quadratic formula. The quadratic formula states that the solutions of a quadratic equation of the form ax2 + bx + c = 0 are given by . A quadratic equation in this form can have no real solutions, one real solution, or two real solutions. 2.4.1: Solving a Linear-Polynomial System of Equations

  15. Common Errors/Misconceptions not identifying all points of intersection incorrectly graphing each equation incorrectly entering the equations into the calculator not understanding that solutions must satisfy every function in a system believing that every system has at least one solution 2.4.1: Solving a Linear-Polynomial System of Equations

  16. Guided Practice Example 2 Use a graph to estimate the real solution(s), if any, to the system of equations . Verify that any identified coordinate pairs are solutions. 2.4.1: Solving a Linear-Polynomial System of Equations

  17. Guided Practice: Example 2, continued Create a graph of the two functions, f(x) and g(x). You can use what you have learned about graphing equations to graph each equation on the same coordinate plane, or you can use a graphing calculator. To graph the system on your graphing calculator, follow the directions appropriate to your calculator modelthat begin on the next slide. 2.4.1: Solving a Linear-Polynomial System of Equations

  18. Guided Practice: Example 2, continued On a TI-83/84: Step 1: Press the [Y=] button. Step 2: Type the first function into Y1, using the [X, T, θ, n] button for the variable x. Press [ENTER]. Step 3: Type the second function into Y2, using the [^] button for powers. Press [ENTER]. Step 4: Press [ZOOM]. Arrow down to 6: ZStandard. Press [ENTER]. 2.4.1: Solving a Linear-Polynomial System of Equations

  19. Guided Practice: Example 2, continued On a TI-Nspire: Step 1: Press the [home] key. Step 2: Arrow over to the graphing icon and press [enter]. Step 3: Type the first function next to f1(x), using the [X] button for the letter x or the [x2] button for a square. Press [enter]. Step 4: To graph the second function, press [menu] and arrow down to 3: Graph Type. Arrow right to bring up the sub-menu, then select 1: Function. Press [enter]. • (continued) 2.4.1: Solving a Linear-Polynomial System of Equations

  20. Guided Practice: Example 2, continued Step 5: At f2(x), type the second function and press [enter]. Step 6: To change the viewing window, press [menu]. Select 4: Window/Zoom and select A: Zoom – Fit. The resulting graph is shown on the next slide. 2.4.1: Solving a Linear-Polynomial System of Equations

  21. Guided Practice: Example 2, continued 2.4.1: Solving a Linear-Polynomial System of Equations

  22. Guided Practice: Example 2, continued Find the coordinates of any apparent intersections. On your graph, estimate where the two equations intersect. It may be necessary to plot additional points. To determine the approximate point(s) of intersection on your graphing calculator, use the following directions. 2.4.1: Solving a Linear-Polynomial System of Equations

  23. Guided Practice: Example 2, continued On a TI-83/84: Step 1: Press [2ND][TRACE] to call up the CALC screen. Arrow down to 5: intersect. Press [ENTER]. Step 2: At the prompt, use the up/down arrow keys to select the Y1 equation, and press [ENTER]. Step 3: At the prompt, use the up/down arrow keys to select the Y2 equation, and press [ENTER]. • (continued) 2.4.1: Solving a Linear-Polynomial System of Equations

  24. Guided Practice: Example 2, continued Step 4: At the prompt, use the arrow keys to move the cursor close to an apparent intersection, and press [ENTER]. The coordinates of the intersection points are displayed. 2.4.1: Solving a Linear-Polynomial System of Equations

  25. Guided Practice: Example 2, continued On a TI-Nspire: Step 1: Press [menu]. Arrow down to 6: Analyze Graph, then arrow right to 4: Intersection. Press [enter]. Step 2: Use the pointing hand to click on each graphed line. The coordinates of the intersection points are displayed. Repeat for each intersection as needed. The points of intersection are (–2, –7), (0, 1), and (2, 9). 2.4.1: Solving a Linear-Polynomial System of Equations

  26. Guided Practice: Example 2, continued Verify that the identified coordinate pairs are solutions to the original system of equations. In order for a coordinate pair to be solution to a system, the coordinates must satisfy both equations of the system. Substitute the coordinates into each of the equations, then evaluate the equations to see if the coordinates result in a true statement. 2.4.1: Solving a Linear-Polynomial System of Equations

  27. Guided Practice: Example 2, continued At the point (–2, –7), let x = –2 and f(x) and g(x) = –7. f(x) = 4x + 1 First equation (–7) = 4(–2) + 1 Substitute –2 for x and –7 for f(x). –7 = –7 g(x) = x3+ 1 Second equation (–7) = (–2)3+ 1 Substitute –2 for x and –7 for g(x). –7 = –7 The identified point (–2, –7) satisfies both f(x) and g(x). 2.4.1: Solving a Linear-Polynomial System of Equations

  28. Guided Practice: Example 2, continued At the point (0, 1), let x = 0 and f(x) and g(x) = 1. f(x) = 4x + 1 First equation = 4(0) + 1 Substitute 0 for x and 1 for f(x). 1 = 1 g(x) = x3+ 1 Second equation = (0)3+ 1 Substitute 0 for x and 1 for g(x). 1 = 1 The identified point (0, 1) satisfies both f(x) and g(x). 2.4.1: Solving a Linear-Polynomial System of Equations

  29. Guided Practice: Example 2, continued At the point (2, 9), let x = 2 and f(x) and g(x) = 9. f(x) = 4x + 1 First equation (9) = 4(2) + 1 Substitute 2 for x and 9 for f(x). 9 = 9 g(x) = x3+ 1 Second equation (9) = (2)3+ 1 Substitute 2 for x and 9 for g(x). 9 = 9 The identified point (2, 9) satisfies both f(x) and g(x). 2.4.1: Solving a Linear-Polynomial System of Equations

  30. Guided Practice: Example 2, continued The solution set to the system of equationsis {(–2, –7), (0, 1), (2, 9)}, as identified on the graph as the intersections of the lines f(x) and g(x). ✔ 2.4.1: Solving a Linear-Polynomial System of Equations

  31. Guided Practice: Example 2, continued 2.4.1: Solving a Linear-Polynomial System of Equations

  32. Guided Practice Example 3 Create a table to approximate the real solution(s), if any, to the system . 2.4.1: Solving a Linear-Polynomial System of Equations

  33. Guided Practice: Example 3, continued Create a table of values for the two functions f(x) and g(x). You can create a table of values by hand similarly to how you have created them in the past, or you can use your graphing calculator. To create a table of values using your graphing calculator, follow the directions appropriate to your calculator model. 2.4.1: Solving a Linear-Polynomial System of Equations

  34. Guided Practice: Example 3, continued On a TI-83/84: Step 1: Press the [Y=] button. Clear any equations from previous calculations. Step 2: Type the first function into Y1, using the [X, T, θ, n] button for the variable x. Press [ENTER]. Step 3: Type the second function into Y2. Use the [^] button for powers. After entering a power, press the right arrow button to get out of exponent mode. Press [ENTER]. • (continued) 2.4.1: Solving a Linear-Polynomial System of Equations

  35. Guided Practice: Example 3, continued Step 4: Press [2ND][WINDOW] to call up the TABLE SETUP screen. Set TblStart to –5 by clearing out any current values and typing [(–)][5][ENTER]. Set ΔTblto 1, then press [2ND][GRAPH] to call up the TABLE screen. A table of values for both equations will be displayed. 2.4.1: Solving a Linear-Polynomial System of Equations

  36. Guided Practice: Example 3, continued On a TI-Nspire: Step 1: Press the [home] key. Step 2: Arrow over to the calculator page, the first icon over, and press [enter]. Step 3: Define f(x) by entering “define” and then the function using the buttons on your keypad. Remember to enter a space after “define”. Press [enter]. • (continued) 2.4.1: Solving a Linear-Polynomial System of Equations

  37. Guided Practice: Example 3, continued Step 4: Define g(x) by entering “define” and then the function using the buttons on your keypad. Remember to press the right arrow key after entering an exponent. Press [enter]. Step 5: Press the [home] key. Arrow over to the spreadsheets page, the fourth icon over, and press [enter]. Step 6: Press [ctrl][T] to switch to a table window. From the list of defined functions that appears, select f(x)f1, and press [enter]. • (continued) 2.4.1: Solving a Linear-Polynomial System of Equations

  38. Guided Practice: Example 3, continued Step 7: Press the right arrow key. From the list of defined functions that appears, select f(x)g1, and press [enter]. A table of values for both equations will be displayed. Both calculators should yield a table that resembles the following. 2.4.1: Solving a Linear-Polynomial System of Equations

  39. Guided Practice: Example 3, continued Estimate the points of intersection. Compare the y-values for the same x-value in your completed table. Look for places where the difference between the values for g(x) and f(x) is decreasing. If the difference between g(x) and f(x) is increasing, then any intersection occurs in the other direction. The point of intersection occurs when f(x) is equal to g(x) or the difference is 0. 2.4.1: Solving a Linear-Polynomial System of Equations

  40. Guided Practice: Example 3, continued Notice that in the table, f(x) equals g(x) when x = –1. Also notice that there appears to be a second intersection point close to the x-values of –3 and –2. To better approximate the second point of intersection, you must find additional values. 2.4.1: Solving a Linear-Polynomial System of Equations

  41. Guided Practice: Example 3, continued Use the following directions to determine additional values on your graphing calculator. On a TI-83/84: Step 1: Press [2ND][WINDOW] to call up the TABLE SETUP screen. Set the TblStart to –3 and ΔTbl to 0.01, clearing out any previous values. Press [ENTER], then press [2ND][GRAPH] to call up the TABLE screen. A table of values for both equations will be displayed. Step 2: Scroll through the table of values to determine the approximate solutions. 2.4.1: Solving a Linear-Polynomial System of Equations

  42. Guided Practice: Example 3, continued On a TI-Nspire: Step 1: Press [menu]. Use the arrow keys to select 5: Table, then 5: Edit Table Settings. Set the Table Start to –3 and Table Step to 0.01. Arrow down to OK and press [enter]. You may need to readjust the Table Step setting to get a better approximation of when f(x) = g(x). 2.4.1: Solving a Linear-Polynomial System of Equations

  43. Guided Practice: Example 3, continued The following table displays information for values of x close to the point we’re interested in: x = –3. Since we know that the difference between g(x) and f(x) is smallest for values of x close to –3, review the table near this value. Determine where the values of g(x) and f(x) have the smallest difference. 2.4.1: Solving a Linear-Polynomial System of Equations

  44. Guided Practice: Example 3, continued According to the generated table of additional values, the value of x with the smallest difference between g(x) and f(x) is –2.96. Because the difference in values is so small and because the coordinates are approximate, it is appropriate to use either the f(x) value or the g(x) value when stating the solution. It can be seen from the tables that –1 and the approximate value of –2.96 are the x-coordinates at which f(x) and g(x) intersect; therefore, the approximate solutions to the system are (–2.96, –0.02) and (–1, –1). 2.4.1: Solving a Linear-Polynomial System of Equations

  45. Guided Practice: Example 3, continued Verify that the identified coordinate pairs are solutions to the original system of equations. In order for a coordinate to be a solution to a system, the coordinate must satisfy both equations of the system. Because our coordinates are estimated, it is quite possible that our results will be nearly equal, but not exactly equal. Substitute the coordinates into each of the equations, then evaluate the equations to see if the coordinates result in a true statement. 2.4.1: Solving a Linear-Polynomial System of Equations

  46. Guided Practice: Example 3, continued At the point (–2.96, –0.02), let x = –2.96 and f(x) and g(x) = –0.02. First equation Substitute –2.96 for x and –0.02 for f(x). –0.02 = –0.02 2.4.1: Solving a Linear-Polynomial System of Equations

  47. Guided Practice: Example 3, continued g(x) = x4 + 3x3 + 1 Second equation (–0.02) = (–2.96)4 + 3(–2.96)3 + 1 Substitute –2.96 for x and –0.02 for g(x). –0.02 ≈ –0.04 The identified point (–2.96, –0.02) nearlysatisfiesbothf(x) and g(x), and canbeconsidered an approximate solution to the system of equations. 2.4.1: Solving a Linear-Polynomial System of Equations

  48. Guided Practice: Example 3, continued At the point (–1, –1), let x = –1 and f(x) and g(x) = –1. First equation Substitute –1 for x and –1 for f(x). –1 = –1 2.4.1: Solving a Linear-Polynomial System of Equations

  49. Guided Practice: Example 3, continued g(x) = x4 + 3x3 + 1 Second equation (–1) = (–1)4 + 3(–1)3 + 1 Substitute –1 for x and –1 for g(x). –1 = –1 The identified point (–1, –1) satisfies both f(x) and g(x), and is a solution to the system of equations. 2.4.1: Solving a Linear-Polynomial System of Equations

  50. Guided Practice: Example 3, continued The approximate solution set to the system is {(–2.96, –0.02), (–1, –1)}, as identified in the tables as the intersection of the equations f(x) and g(x). 2.4.1: Solving a Linear-Polynomial System of Equations

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