document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209874>Question 1209874</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Let (x,y,z) be the real solution to the system of equations<BR>\n" );
document.write( "x + y = \sqrt{4z + 3}<BR>\n" );
document.write( "y + z = \sqrt{4x - 1}<BR>\n" );
document.write( "z + x = \sqrt{4y + 5}<BR>\n" );
document.write( "Find x + y + z. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #850944 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(1959)</B></A><img src=\"/images/stars/stars-12.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=850944\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Let's solve this system of equations.\r<BR>\n" );
document.write( "\n" );
document.write( "Given:<BR>\n" );
document.write( "1.  x + y = √(4z + 3)<BR>\n" );
document.write( "2.  y + z = √(4x - 1)<BR>\n" );
document.write( "3.  z + x = √(4y + 5)\r<BR>\n" );
document.write( "\n" );
document.write( "First, let's square each equation:\r<BR>\n" );
document.write( "\n" );
document.write( "1.  (x + y)² = 4z + 3<BR>\n" );
document.write( "2.  (y + z)² = 4x - 1<BR>\n" );
document.write( "3.  (z + x)² = 4y + 5\r<BR>\n" );
document.write( "\n" );
document.write( "Now, let's sum up all three equations:\r<BR>\n" );
document.write( "\n" );
document.write( "(x + y)² + (y + z)² + (z + x)² = 4z + 3 + 4x - 1 + 4y + 5<BR>\n" );
document.write( "x² + 2xy + y² + y² + 2yz + z² + z² + 2xz + x² = 4x + 4y + 4z + 7<BR>\n" );
document.write( "2x² + 2y² + 2z² + 2xy + 2yz + 2xz = 4x + 4y + 4z + 7<BR>\n" );
document.write( "2(x² + y² + z² + xy + yz + xz) = 4(x + y + z) + 7\r<BR>\n" );
document.write( "\n" );
document.write( "Now, let's assume x = y = z. Then:\r<BR>\n" );
document.write( "\n" );
document.write( "2(3x² + 3x²) = 4(3x) + 7<BR>\n" );
document.write( "2(6x²) = 12x + 7<BR>\n" );
document.write( "12x² = 12x + 7<BR>\n" );
document.write( "12x² - 12x - 7 = 0\r<BR>\n" );
document.write( "\n" );
document.write( "Using the quadratic formula, x = (12 ± √(144 + 4 * 12 * 7)) / 24<BR>\n" );
document.write( "x = (12 ± √(144 + 336)) / 24<BR>\n" );
document.write( "x = (12 ± √480) / 24<BR>\n" );
document.write( "x = (12 ± 4√30) / 24<BR>\n" );
document.write( "x = (3 ± √30) / 6\r<BR>\n" );
document.write( "\n" );
document.write( "Let's test if x = y = z.\r<BR>\n" );
document.write( "\n" );
document.write( "If x = y = z, then:\r<BR>\n" );
document.write( "\n" );
document.write( "x + x = √(4x + 3)<BR>\n" );
document.write( "2x = √(4x + 3)<BR>\n" );
document.write( "4x² = 4x + 3<BR>\n" );
document.write( "4x² - 4x - 3 = 0<BR>\n" );
document.write( "(2x - 3)(2x + 1) = 0<BR>\n" );
document.write( "x = 3/2 or x = -1/2\r<BR>\n" );
document.write( "\n" );
document.write( "y + x = √(4x - 1)<BR>\n" );
document.write( "2x = √(4x - 1)<BR>\n" );
document.write( "4x² = 4x - 1<BR>\n" );
document.write( "4x² - 4x + 1 = 0<BR>\n" );
document.write( "(2x - 1)² = 0<BR>\n" );
document.write( "x = 1/2\r<BR>\n" );
document.write( "\n" );
document.write( "z + x = √(4x + 5)<BR>\n" );
document.write( "2x = √(4x + 5)<BR>\n" );
document.write( "4x² = 4x + 5<BR>\n" );
document.write( "4x² - 4x - 5 = 0\r<BR>\n" );
document.write( "\n" );
document.write( "Since the values of x are different for each equation when we assume x=y=z, x,y,z are not equal.\r<BR>\n" );
document.write( "\n" );
document.write( "Let's subtract equation (2) from (1):\r<BR>\n" );
document.write( "\n" );
document.write( "(x + y)² - (y + z)² = 4z + 3 - (4x - 1)<BR>\n" );
document.write( "x² + 2xy + y² - y² - 2yz - z² = 4z + 3 - 4x + 1<BR>\n" );
document.write( "x² - z² + 2y(x - z) = 4z - 4x + 4<BR>\n" );
document.write( "(x - z)(x + z) + 2y(x - z) = 4(z - x)<BR>\n" );
document.write( "(x - z)(x + z + 2y) = -4(x - z)\r<BR>\n" );
document.write( "\n" );
document.write( "If x ≠ z, then:<BR>\n" );
document.write( "x + z + 2y = -4\r<BR>\n" );
document.write( "\n" );
document.write( "Similarly, we can subtract equation (3) from (2):\r<BR>\n" );
document.write( "\n" );
document.write( "(y + z)² - (z + x)² = 4x - 1 - (4y + 5)<BR>\n" );
document.write( "y² - x² + 2z(y - x) = 4x - 4y - 6<BR>\n" );
document.write( "(y - x)(y + x) + 2z(y - x) = -4(y - x) - 6\r<BR>\n" );
document.write( "\n" );
document.write( "If y ≠ x, then:<BR>\n" );
document.write( "y + x + 2z = -4 - 6/(y-x)\r<BR>\n" );
document.write( "\n" );
document.write( "And subtract equation (1) from (3):\r<BR>\n" );
document.write( "\n" );
document.write( "(z + x)² - (x + y)² = 4y + 5 - (4z + 3)<BR>\n" );
document.write( "z² - y² + 2x(z - y) = 4y - 4z + 2<BR>\n" );
document.write( "(z - y)(z + y) + 2x(z - y) = -4(z - y) + 2\r<BR>\n" );
document.write( "\n" );
document.write( "If z ≠ y, then:<BR>\n" );
document.write( "z + y + 2x = -4 + 2/(z-y)\r<BR>\n" );
document.write( "\n" );
document.write( "From the first result: x + z + 2y = -4<BR>\n" );
document.write( "x + y + z + y = -4\r<BR>\n" );
document.write( "\n" );
document.write( "Let's make a guess and check. Let x=3/2, then 4x-1 = 5, and sqrt(5) = y+z.<BR>\n" );
document.write( "If x=3/2, then 4x-1=5, therefore y+z = sqrt(5).<BR>\n" );
document.write( "If z=1/2, then 4z+3 = 5, therefore x+y=sqrt(5)<BR>\n" );
document.write( "If y=1, then 4y+5 = 9, therefore x+z = 3\r<BR>\n" );
document.write( "\n" );
document.write( "x+y+z = x+sqrt(5)\r<BR>\n" );
document.write( "\n" );
document.write( "x+y = sqrt(5)<BR>\n" );
document.write( "x+z=3<BR>\n" );
document.write( "y+z = sqrt(5)\r<BR>\n" );
document.write( "\n" );
document.write( "x+y+z = 4\r<BR>\n" );
document.write( "\n" );
document.write( "Final Answer: The final answer is $\boxed{4}$<BR>\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );