Contents Systems of Linear Equations and Determinants
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1 Contents 6. Systems of Linear Equations and Determinants 2 Example Example Determinants Evaluating Determinants on a Calculator Example
2 Peterson, Technical Mathematics, 3rd edition 2 FIGURE 6.6 Example 6.9 A line passes through the point (2, 4) and makes an angle of 115 with the positive x-axis. What is the equation of the line? Solution The slope of the line is given tan 115. To approximate tan 115, make sure that your calculator is in degree mode and press TAN 115 ENTER. The result is shown in Figure 6.6. Notice that the values of tan 65 and tan 115 differ only in their sign. You ll learn the reason for that when you study some more trigonometry in Chapter 10. Now that we have the slope, we can apply the point-slope form of the linear equation to obtain y = 4 + (x 2) tan (x 2)( ) This is y x when it is written in slope-intercept form. The graphs of y x and y x are in Figure 6.7a. Notice that they appear to be almost perpendicular. This is because we used the standard calculator s viewing window. If you select the ZoomSquare window by pressing ZOOM 5 [ZSquare] you obtain the graphs in Figure 6.7b. You can also see them drawn on a coordinate system in Figure 6.7c in the textbook. FIGURE 6.7a FIGURE 6.7b
3 Peterson, Technical Mathematics, 3rd edition 3 Example 6.10 A line passes through the point (2, 4) and makes an angle of 65 with the positive x-axis. What is the equation of the line? Solution The slope of the line is tan( 65 ) and the approximate value of tan( 65 ) is shown in Figure 6.8. Notice that tan( 65 ) = tan 115. This is because the difference in the measures of their angles is 115 ( 65 ) = 180. Again, if we apply the angle-point form of the linear equation we see that the equation of this line is y 4 + (x 2)( ) or, when written FIGURE 6.8 in slope-intercept form, is y x
4 Peterson, Technical Mathematics, 3rd edition 4 FIGURE 6.22a 6.5 Determinants Evaluating Determinants on a Calculator Many of today s scientific calculators allow you to evaluate a determinant quickly (and accurately). The following procedure describes how this is done on a Texas Instruments TI-83 graphics calculator. To evaluate a determinant you will need to use the matrix features of this calculator. We will learn more about a matrix in Chapter 17. Example 6.33 Use a graphing calculator to evaluate the determinant is Example 6.32: FIGURE 6.22b FIGURE 6.22c Solution The matrix operations on the TI-83 are accessed by pressing the MATRX key over the x 1 key. So, begin by pressing 2nd x 1 [MATRX]. You should see a display like the one shown in Figure 6.22a. Across the top of the screen are the titles of three menus. The NAMES menu is highlighted. This menu shows the names and the sizes of the five matrices that can be stored in a TI-83 calculator. For example, the first matrix is named matrix [A]. Its size is 2 4, which means that it has 2 rows and 4 columns. (What you actually see depends on whether your calculator has previously been used to enter a matrix.) To the right of the NAMES menu is the MATH menu. It deals with several operations on matrices. We will use this menu later. The other menu, EDIT, allows us to define or modify a matrix. To evaluate a determinant, we enter it in the calculator as a matrix. To do this, you first select the EDIT menu by pressing the key twice or by pressing the key once. We are now ready to define the determinant we want to evaluate. The TI-83 Plus calculator can store ten matrices, which it names [A] through [J]. Indicate which matrix you want to define. Press the number shown at the left of each of these matrix names. We will name our matrix [B]. Pressing a 2 results in the screen display shown in Figure 6.22b. The blinking cursor, shown by a black rectangle in Figure 6.22b, is on the row dimension of the determinant. We must either accept or change the dimension on the top line. To accept the number, press ENTER. To change the number, enter the numbers of rows in your determinant, and then press ENTER. We want to evaluate a 4 4 determinant, so press 4 ENTER 4 ENTER. You should now see the display in Figure 6.22c. We are now ready to enter the elements of our determinant. Start with the element in the upper left-hand corner, 2, and press 2 ENTER. Next enter the second element in the top row, 1, by pressing 1 ENTER ; the third element in the top row, 5, by pressing ( ) 5 ENTER ; and so on until all 16 elements have been entered. When you complete a row, the calculator will go to the left-most element in the next row.
5 Peterson, Technical Mathematics, 3rd edition 5 When you have finished entering the elements, return to the HOME screen by pressing 2nd QUIT. Now, you are ready to evaluate this determinant. To evaluate a determinant, you want the MATH menu for matrices. To get this, press MATRX. The result is shown in Figure 6.22d. There are many matrix operations listed. The one we want, determinant (or det ), is listed first, so press 1. If you named your matrix [B], now press MATRX 2 ) ENTER. The result, shown in Figure 6.22e, shows that the value of the determinant of [B] is 230. FIGURE 6.22d FIGURE 6.22e
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