document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1209658>Question 1209658</A>: <I><SPAN STYLE=\"word-break: break-all;\"> The real numbers x and y satisfy<BR>\n" );
document.write( "x^2 + y^2 - 8x + 6y + 23 = 0.<BR>\n" );
document.write( "Find the largest possible value of x + y. </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849769 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=849769\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to find the largest possible value of x + y:\r<BR>\n" );
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document.write( "1. **Rewrite the equation:**\r<BR>\n" );
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document.write( "The given equation represents a circle. We can rewrite it in standard form by completing the square for both x and y:\r<BR>\n" );
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document.write( "x² - 8x + y² + 6y + 23 = 0<BR>\n" );
document.write( "(x² - 8x + 16) + (y² + 6y + 9) + 23 - 16 - 9 = 0<BR>\n" );
document.write( "(x - 4)² + (y + 3)² = 2\r<BR>\n" );
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document.write( "2. **Geometric Interpretation:**\r<BR>\n" );
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document.write( "This equation represents a circle with center (4, -3) and radius √2.\r<BR>\n" );
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document.write( "3. **Express y in terms of x:**\r<BR>\n" );
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document.write( "We want to maximize x + y. Let's express y in terms of x using the equation of the circle:\r<BR>\n" );
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document.write( "(y + 3)² = 2 - (x - 4)²<BR>\n" );
document.write( "y + 3 = ±√[2 - (x - 4)²]<BR>\n" );
document.write( "y = -3 ± √[2 - (x - 4)²]\r<BR>\n" );
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document.write( "4. **Maximize x + y:**\r<BR>\n" );
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document.write( "We want to maximize the function f(x) = x + y. Substituting the expression for y, we get:\r<BR>\n" );
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document.write( "f(x) = x - 3 ± √[2 - (x - 4)²]\r<BR>\n" );
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document.write( "To find the maximum value of f(x), we can consider the line x + y = k, where k is a constant. We want to find the largest value of k such that the line intersects the circle.  Geometrically, this line will be tangent to the circle at the point that maximizes x + y.\r<BR>\n" );
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document.write( "The line x + y = k can be written as y = -x + k. The slope of this line is -1.\r<BR>\n" );
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document.write( "5. **Tangent Line:**\r<BR>\n" );
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document.write( "The line connecting the center of the circle (4, -3) to the point of tangency will be perpendicular to the tangent line x + y = k. Thus, the slope of this line is 1. Let (x,y) be the point of tangency.<BR>\n" );
document.write( "(y - (-3))/(x - 4) = 1<BR>\n" );
document.write( "y + 3 = x - 4<BR>\n" );
document.write( "y = x - 7\r<BR>\n" );
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document.write( "Substitute into the equation for the circle:<BR>\n" );
document.write( "(x-4)^2 + (x - 7 + 3)^2 = 2<BR>\n" );
document.write( "(x-4)^2 + (x-4)^2 = 2<BR>\n" );
document.write( "2(x-4)^2 = 2<BR>\n" );
document.write( "(x-4)^2 = 1<BR>\n" );
document.write( "x - 4 = ±1<BR>\n" );
document.write( "x = 5 or x = 3\r<BR>\n" );
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document.write( "If x = 5, y = 5 - 7 = -2, so x + y = 3<BR>\n" );
document.write( "If x = 3, y = 3 - 7 = -4, so x + y = -1\r<BR>\n" );
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document.write( "The maximum value of x+y is 3.\r<BR>\n" );
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document.write( "**Alternatively:**<BR>\n" );
document.write( "The maximum value of x + y occurs when the line x + y = k is tangent to the circle. The distance from the center of the circle to the line must be equal to the radius.\r<BR>\n" );
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document.write( "|4 + (-3) - k| / sqrt(1^2 + 1^2) = sqrt(2)<BR>\n" );
document.write( "|1 - k| / sqrt(2) = sqrt(2)<BR>\n" );
document.write( "|1 - k| = 2<BR>\n" );
document.write( "1 - k = 2 or 1 - k = -2<BR>\n" );
document.write( "k = -1 or k = 3.\r<BR>\n" );
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document.write( "The largest value is 3.\r<BR>\n" );
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document.write( "Final Answer: The final answer is $\boxed{3}$<BR>\n" );
document.write( " </SPAN>\n" );
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