document.write( "Question 1152633: Solve, Conjugate(z) - 12/z = 1-i, expressing your answer in the form a + ib where a and b are real numbers. [4 marks] \n" ); document.write( "
Algebra.Com's Answer #774740 by Edwin McCravy(20056)\"\" \"About 
You can put this solution on YOUR website!
\r\n" );
document.write( "\"Conjugate%28z%29+-+12%2Fz+=+1-i\"\r\n" );
document.write( "\r\n" );
document.write( "Let z = x+yi, where x and y are real numbers\r\n" );
document.write( "\r\n" );
document.write( "\"Conjugate%28x%2Byi%29+-+12%2F%28x%2Byi%29+=+1-i\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-yi%29+-+%2812%2F%28x%2Byi%29%29%28%28x-yi%29%2F%28x-yi%29%29+=+1-i\"\r\n" );
document.write( "\r\n" );
document.write( "\"x-yi+-+%2812x-12yi%5E%22%22%29%2F%28x%5E2-y%5E2i%5E2%29+=+1-i\"\r\n" );
document.write( "\r\n" );
document.write( "i² = -1\r\n" );
document.write( "\r\n" );
document.write( "\"x-yi+-+%2812x-12yi%5E%22%22%29%2F%28x%5E2-y%5E2%28-1%29%29+=+1-i\"\r\n" );
document.write( "\r\n" );
document.write( "\"x-yi+-+%2812x-12yi%5E%22%22%29%2F%28x%5E2%2By%5E2%29+=+1-i\"\r\n" );
document.write( "\r\n" );
document.write( "Put parentheses around the fraction term:\r\n" );
document.write( "\r\n" );
document.write( "\"x-yi+-+%28%2812x-12yi%5E%22%22%29%2F%28x%5E2%2By%5E2%29%29+=+1-i\"\r\n" );
document.write( "\r\n" );
document.write( "Break the fraction term into two fractions:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "Remove the parentheses:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "In an equation with both real and imaginary terms, the real\r\n" );
document.write( "and imaginary terms are like oil and water. The real terms\r\n" );
document.write( "on the left equals the real terms on the right, and the \r\n" );
document.write( "imaginary terms on the left equal the imaginary terms on the\r\n" );
document.write( "right.  So we split it into two equations.\r\n" );
document.write( "\r\n" );
document.write( "The REAL equation is:\r\n" );
document.write( "\r\n" );
document.write( "\"x-+%2812x%5E%22%22%29%2F%28x%5E2%2By%5E2%29+=+1\"\r\n" );
document.write( "\r\n" );
document.write( "The IMAGINARY equation is\r\n" );
document.write( "\r\n" );
document.write( "\"-yi%2B%2812yi%5E%22%22%29%2F%28x%5E2%2By%5E2%29+=+-i\"\r\n" );
document.write( "\r\n" );
document.write( "which we divide through by -i\r\n" );
document.write( "\r\n" );
document.write( "\"y-%2812y%5E%22%22%29%2F%28x%5E2%2By%5E2%29+=+1\"\r\n" );
document.write( "\r\n" );
document.write( "So we have the system of equations:\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "\r\n" );
document.write( "The equations are symmetrical in x and y.  So we can substitute \r\n" );
document.write( "x for y.  \r\n" );
document.write( "\r\n" );
document.write( "\"x-+%2812x%5E%22%22%29%2F%28x%5E2%2By%5E2%29+=+1\"\r\n" );
document.write( "\r\n" );
document.write( "Replace y by x:\r\n" );
document.write( "\r\n" );
document.write( "\"x-+%2812x%5E%22%22%29%2F%28x%5E2%2Bx%5E2%29+=+1\"\r\n" );
document.write( "\r\n" );
document.write( "\"x-+%2812x%5E%22%22%29%2F%282x%5E2%29+=+1\"\r\n" );
document.write( "\r\n" );
document.write( "\"x-+6%2Fx+=+1\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2-6+=+x\"\r\n" );
document.write( "\r\n" );
document.write( "\"x%5E2-x-6=0\"\r\n" );
document.write( "\r\n" );
document.write( "\"%28x-3%29%28x%2B2%29=0\"\r\n" );
document.write( "\r\n" );
document.write( "x - 3 = 0; x + 2 = 0\r\n" );
document.write( "    x = 3;     x = -2\r\n" );
document.write( "\r\n" );
document.write( "So there are two solutions: (x,y) = (3,3), (x,y) = (-2,-2)\r\n" );
document.write( "\r\n" );
document.write( "Expressing in a+bi form:\r\n" );
document.write( "\r\n" );
document.write( "z = x+yi = 3+3i   and  z = x+yi = -2-2i\r\n" );
document.write( "\r\n" );
document.write( "Edwin
\n" ); document.write( "
\n" );