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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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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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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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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