Solution

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Let s tackle the second equation and get it into a form that expresses y in terms of x First, we can add 8x to each side, getting 2y = 8x + 4 When we divide through by 2, we get y = 4x + 2

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Rename and Replace

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Now we can substitute (4x + 2) for y in the first original equation, obtaining 3x (4x + 2) = 1 When we apply the distributive law on the left side of the equals sign, we get 3x (4 x + 2 ) = 1 This is the equivalent of 3x + [ 1(4 x + 2 )] = 1 which simplifies to 3x 4 x 2 = 1 When we add 2 to each side, we get 3x 4 x = 1 + 2 We can use the distributive law backward to morph the left side of this equation, obtaining (3 4 )x = 1 + 2 Now we can divide through by (3 4 ) to get x = ( 1 + 2 )/(3 4 )

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It s a mess, all right! But we ve found a real number that s equal to x We can plug this number into the SI equation we derived earlier, getting y = 4[( 1 + 2 )/(3 4 )] + 2 = ( 4 + 8 )/(3 4 ) + 2 Believe it or not, this can be simplified But we must take a step back, and then we can take two steps forward Let s complexify the number 2 and write it as twice the denominator in the fraction above, divided by that denominator The idea is to get a common denominator, add some fractions, and get a simpler expression as a result In mathematical terms, 2 = 2(3 4 )/(3 4 ) It takes some intuition to see, in advance, how a scheme like this will work (With practice, you ll develop this sixth sense ) Our solution for y can now be rewritten as y = ( 4 + 8 )/(3 4 ) + 2(3 4 )/(3 4 ) This gives us a sum of two fractions with the common denominator (3 4 ) Therefore: y = [( 4 + 8 ) + 2(3 4 )]/(3 4 )

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262 Two-by-Two Linear Systems

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Applying the distributive law, we get y = ( 4 + 8 + 6 8 )/(3 4 ) When we add up the terms in the numerator here, we get our reward: y = 2/(3 4 ) We ve arrived at our solutions! They are: x = ( 1 + 2 )/(3 4 ) and y = 2/(3 4 )

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How about some extra credit

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Plug the above numbers into the original equations for x and y, and verify that the answers we got are correct You re on your own! Here s a hint: (3 4 ) divided by itself is equal to 1

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Practice Exercises

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This is an open-book quiz You may (and should) refer to the text as you solve these problems Don t hurry! You ll find worked-out answers in App B The solutions in the appendix may not represent the only way a problem can be figured out If you think you can solve a particular problem in a quicker or better way than you see there, by all means try it! 1 The sum of two numbers is 44 Their difference is 10 What are the two numbers Use the morph-and-mix method 2 The sum of two numbers is 100 One of them is 6 times the other What are the two numbers Use the morph-and-mix method 3 Imagine that you and I are traveling in a car on a level highway at constant speed I m the driver There are no other vehicles or living things in sight You throw a baseball straight out in front of the car The ball strikes the pavement at 135 miles per hour (mi/h) Then you throw another baseball directly backward, exactly as hard as the first one The second ball hits the highway, moving opposite to the direction of the car, at 15 mi/h (The ball is not only moving backward relative to the car; it s also moving backward relative to the pavement!) How fast am I driving How fast do you hurl the baseballs relative to the car Forget about the possible effects of wind and gravity Here s a hint: Feel free to draw diagrams 4 The sum of two numbers is 83 Their difference is 13 What are the two numbers Use double elimination