1.3 Applications of Systems of Linear Equations

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1 .3 Applications of Sstems of Linear Equations 5.3 Applications of Sstems of Linear Equations ( n, n ) ( 3, 3 ) (, ) (, ) Polnomial Curve Fitting Figure. Set up and solve a sstem of equations to fit a polnomial function to a set of data points. Set up and solve a sstem of equations to represent a network. Sstems of linear equations arise in a wide variet of applications. In this section ou will look at two applications, and ou will see more in subsequent chapters. The first application shows how to fit a polnomial function to a set of data points in the plane. The second application focuses on networks and Kirchhoff s Laws for electricit. POLYNOMIAL CURVE FITTING Suppose n points in the -plane,,,,..., n, n represent a collection of data and ou are asked to find a polnomial function of degree n p a a a... a n n whose graph passes through the specified points. This procedure is called polnomial curve fitting. If all -coordinates of the points are distinct, then there is precisel one polnomial function of degree n (or less) that fits the n points, as shown in Figure.. To solve for the n coefficients of p, substitute each of the n points into the polnomial function and obtain n linear equations in n variables a, a, a,..., a. n a a a... n a n a a a... n a n. 8 6 Figure.5 (, ) (, ) (3, ) p() 3 Simulation Eplore this concept further with an electronic simulation available at a a n a n... a n n n n Eample demonstrates this procedure with a second-degree polnomial. Polnomial Curve Fitting Determine the polnomial p a a a whose graph passes through the points,,,, and 3,. Substituting,, and 3 into p and equating the results to the respective -values produces the sstem of linear equations in the variables a, a, and a shown below. p a a a a a a p a a a a a a p3 a a 3 a 3 a 3a 9a The solution of this sstem is a, a 8, and a 8 so the polnomial function is p 8 8. Figure.5 shows the graph of p.

2 6 Chapter Sstems of Linear Equations (, ) p() 8 (, 5) (, 3) (, ) (, ) 3 Figure.6 Find a polnomial that fits the points Polnomial Curve Fitting, 3,, 5,,,,, and,. Because ou are given five points, choose a fourth-degree polnomial function p a a a a 3 3 a. Substituting the given points into p equations. a a a 8a 3 6a 3 a a a a 3 a 5 a a a a a 3 a a a a 8a 3 6a The solution of these equations is a, a 3, a, which means the polnomial function is p Figure.6 shows the graph of p. produces the following sstem of linear a 3 8, a 7 The sstem of linear equations in Eample is relativel eas to solve because the -values are small. For a set of points with large -values, it is usuall best to translate the values before attempting the curve-fitting procedure. The net eample demonstrates this approach. Translating Large -Values Before Curve Fitting Find a polnomial that fits the points,, 3, 3, 5, 5 6, 3, Because the given -values are large, use the translation z 8 to obtain z, z, z 3, 3 z, z 5, 5, 3, 7, 5,, 5, This is the same set of points as in Eample. So, the polnomial that fits these points is pz 3z z 8z 3 7z z 5 z 3 z 3 7 z. Letting z 8, ou have 8,,,, 9,,,,,.,. p

3 .3 Applications of Sstems of Linear Equations 7 An Application of Curve Fitting Find a polnomial that relates the periods of the three planets that are closest to the Sun to their mean distances from the Sun, as shown in the table. Then test the accurac of the fit b using the polnomial to calculate the period of Mars. (In the table, the mean distance is given in astronomical units, and the period is given in ears.) Planet Mercur Venus Earth Mars Mean Distance Period Begin b fitting a quadratic polnomial function p a a a to the points.387,.,.73,.65, and,. The sstem of linear equations obtained b substituting these points into p is a.387a.387 a. a.73a.73 a.65 a a a. The approimate solution of the sstem is a.63, which means that an approimation of the polnomial function is p Using p to evaluate the period of Mars produces p.5.98 ears. Note that the actual period of Mars is.88 ears. Figure.7 compares the estimate with the actual period graphicall. a.69, a.55 Period (in ears) Mercur (.387,.) Mean distance from the Sun (in astronomical units) Figure.7 (.5,.88) = p() Venus Earth Mars (.,.) (.73,.65)

4 8 Chapter Sstems of Linear Equations As illustrated in Eample, a polnomial that fits some of the points in a data set is not necessaril an accurate model for other points in the data set. Generall, the farther the other points are from those used to fit the polnomial, the worse the fit. For instance, the mean distance of Jupiter from the Sun is 5.3 astronomical units. Using p in Eample to approimate the period gives 5.33 ears a poor estimate of Jupiter s actual period of.86 ears. The problem of curve fitting can be difficult. Tpes of functions other than polnomial functions ma provide better fits. For instance, look again at the curve-fitting problem in Eample. Taking the natural logarithms of the given distances and periods produces the following results. Planet Mercur Venus Earth Mars Mean Distance ln Period ln Now, fitting a polnomial to the logarithms of the distances and periods produces the linear relationship ln 3 ln shown in Figure.8. ln Earth Venus Mars ln = 3 ln ln Mercur Figure.8 From ln 3 ln, it follows that 3, or 3. In other words, the square of the period (in ears) of each planet is equal to the cube of its mean distance (in astronomical units) from the Sun. Johannes Kepler first discovered this relationship in 69. LINEAR ALGEBRA APPLIED Researchers in Ital studing the acoustical noise levels from vehicular traffic at a bus three-wa intersection on a college campus used a sstem of linear equations to model the traffic flow at the intersection. To help formulate the sstem of equations, operators stationed themselves at various locations along the intersection and counted the numbers of vehicles going b. (Source: Acoustical Noise Analsis in Road Intersections: A Case Stud, Guarnaccia, Claudio, Recent Advances in Acoustics & Music, Proceedings of the th WSEAS International Conference on Acoustics & Music: Theor & Applications, June, ) gemphotograph/shutterstock.com

5 NETWORK ANALYSIS Networks composed of branches and junctions are used as models in such fields as economics, traffic analsis, and electrical engineering. In a network model, ou assume that the total flow into a junction is equal to the total flow out of the junction. For instance, the junction shown in Figure.9 has 5 units flowing into it, so there must be 5 units flowing out of it. You can represent this with the linear equation 5..3 Applications of Sstems of Linear Equations 9 5 Figure.9 Because each junction in a network gives rise to a linear equation, ou can analze the flow through a network composed of several junctions b solving a sstem of linear equations. Eample 5 illustrates this procedure. Analsis of a Network 3 Figure Set up a sstem of linear equations to represent the network shown in Figure.. Then solve the sstem. Each of the network s five junctions gives rise to a linear equation, as follows. The augmented matri for this sstem is Gauss-Jordan elimination produces the matri From the matri above, ou can see that 5, Letting t 5, ou have t, , t 3,. 3. Junction Junction Junction 3 Junction Junction 5 3 5, 3 t, where t is an real number, so this sstem has infinitel man solutions. and t, 5. 5 t

6 3 Chapter Sstems of Linear Equations REMARK A closed path is a sequence of branches such that the beginning point of the first branch coincides with the end point of the last branch. In Eample 5, suppose ou could control the amount of flow along the branch labeled 5. Using the solution of Eample 5, ou could then control the flow represented b each of the other variables. For instance, letting t would reduce the flow of and 3 to zero, as shown in Figure.. You ma be able to see how the tpe of network analsis demonstrated in Eample 5 could be used in problems dealing with the flow of traffic through the streets of a cit or the flow of water through an irrigation sstem. An electrical network is another tpe of network where analsis is commonl applied. An analsis of such a sstem uses two properties of electrical networks known as Kirchhoff s Laws.. All the current flowing into a junction must flow out of it.. The sum of the products IR ( I is current and R is resistance) around a closed path is equal to the total voltage in the path. In an electrical network, current is measured in amperes, or amps A, resistance is measured in ohms, and the product of current and resistance is measured in volts V. The smbol represents a batter. The larger vertical bar denotes where the current flows out of the terminal. The smbol denotes resistance. An arrow in the branch indicates the direction of the current. 3 Figure. 5 Analsis of an Electrical Network I R = 3Ω I R = Ω Path 7 V Path R 3 = Ω 8 V Figure. Determine the currents I, I, and for the electrical network shown in Figure.. Appling Kirchhoff s first law to either junction produces I I Junction or Junction and appling Kirchhoff s second law to the two paths produces R I R I 3I I 7 R I R 3 I 8. Path Path So, ou have the following sstem of three linear equations in the variables I, I, and. I I 3I I 7 I 8 Appling Gauss-Jordan elimination to the augmented matri produces the reduced row-echelon form which means I amp, I amps, and amp.

7 .3 Applications of Sstems of Linear Equations 3 Analsis of an Electrical Network Determine the currents I, I,, I, I 5, and I 6 for the electrical network shown in Figure.3. 7 V R = Ω I V V I Figure.3 Path Path Path 3 I R = Ω I 5 I 6 R = Ω R R 3 = 5 = Ω Ω 3 R 6 = Ω Appling Kirchhoff s first law to the four junctions produces I I I I I I 6 I 5 I I 6 I 5 Junction Junction Junction 3 Junction and appling Kirchhoff s second law to the three paths produces I I I I I 5 7 I 5 I 6. Path Path Path 3 You now have the following sstem of seven linear equations in the variables I, I,, I, I 5, and. I I The augmented matri for this sstem is I 6 Using Gauss-Jordan elimination, a graphing utilit, or a software program, solve this sstem to obtain I, I I I I I I, I I I, I 5 I 5 I 5 I 5 I 6 I 6 I 6 I, 7. 7 I 5 3, meaning I amp, I amps, amp, I amp, I 5 3 amps, and I 6 amps. and I 6