Algebra Know-It-ALL
Solve the Two-by-Two Let’s continue our quest to solve the three-by-three linear system. We have obtained a two-by- two system i ...
286 Larger Linear Systems Adding these equations, we get 410 x− 697 y=− 328 − 187 x+ 697 y= 119 223 x=− 209 Dividing through by ...
10 x= 17 y− 8 x= (17/10)y− 8/10 The second equation morphs as follows: − 11 x+ 41 y= 7 − 11 x=− 41 y+ 7 11 x= 41 y− 7 x= (41/11) ...
288 Larger Linear Systems It agrees again! This practically guarantees that our answers are correct. We’ve found x and y by two ...
Here’s a challenge! In a problem as messy and lengthy as this, it’s vital to check the derived solutions. Here are the original ...
290 Larger Linear Systems General Linear Systems Now that you’ve seen two-by-two and three-by-three linear systems, you might wo ...
+x +y +z x –y z Figure 18-1 Cartesian three-space. The x axis increases positively from left to right, the y axis increases ...
292 Larger Linear Systems Three different planes, such that each pair of planes intersects in a different line, and all three o ...
Things get more interesting with two-by-three linear systems (two equations and three variables x,y, and z). The graph appears a ...
294 Larger Linear Systems Practice Exercises This is an open-book quiz. You may (and should) refer to the text as you solve thes ...
Derive a two-by-two linear system in x and y from the solution to Prob. 1, along with the equation in x and y that we derived f ...
296 CHAPTER 19 The Matrix Morphing Game There’s a neat way to solve three-by-three linear systems by removing the variables from ...
The matrix form Once we have morphed all the equations into the proper form, we can remove the variables and equals signs. Then ...
298 The Matrix Morphing Game Now we have these three equations that make up the linear system: 0 x+ 7 y− 3 z= 3 2 x+ 0 y+ 8 z=− ...
Multiply We may multiply all the elements in any row by a nonzero constant, keeping the elements in the same order from left to ...
300 The Matrix Morphing Game The final goal The matrix morphing game, like any sensible game, has an ultimate objective. Our goa ...
Then we make the matrix: a 1 b 1 c 1 d 1 a 2 b 2 c 2 d 2 a 3 b 3 c 3 d 3 Next, we get the matrix into echelon form, which looks ...
302 The Matrix Morphing Game Formatting the equations None of these equations is in the proper form for assembling a matrix. Let ...
Now let’s get 0 at the extreme left in the second row. If we multiply the second row by −3/5, we come up with 3 − 2 111 − 3 12 6 ...
304 The Matrix Morphing Game Let’s make the number 133 vanish. We can use the third row as the “weapon” here. If we multiply the ...
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