document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1186000>Question 1186000</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Solve for the unknown calories using the concept of linear equations<BR>\n" );
document.write( "1.) 126x + 198y = 150<BR>\n" );
document.write( "2.) 338x + 43y = 303<BR>\n" );
document.write( " With graphing method and subtitution, elimination, determinants </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #849498 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=849498\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> Here's how to solve the system of equations using the methods you requested:\r<BR>\n" );
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document.write( "**1. Graphing Method:**\r<BR>\n" );
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document.write( "1.  **Rewrite the equations in slope-intercept form (y = mx + b):**<BR>\n" );
document.write( "    *   126x + 198y = 150  =>  y = (-126/198)x + (150/198)  => y ≈ -0.636x + 0.758<BR>\n" );
document.write( "    *   338x + 43y = 303  =>  y = (-338/43)x + (303/43)  => y ≈ -7.86x + 7.047\r<BR>\n" );
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document.write( "2.  **Plot the lines:** Graph both equations on the same coordinate plane.  The point where the lines intersect is the solution.\r<BR>\n" );
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document.write( "3.  **Approximate the solution:** From the graph, you'll see that the lines intersect somewhere around x ≈ 0.8 and y ≈ 0.3.  Graphing isn't always perfectly precise, so these are just estimates.\r<BR>\n" );
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document.write( "**2. Substitution Method:**\r<BR>\n" );
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document.write( "1.  **Solve one equation for one variable:**  Let's solve the first equation for y:<BR>\n" );
document.write( "    y = (-126/198)x + (150/198)\r<BR>\n" );
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document.write( "2.  **Substitute:** Substitute this expression for y into the second equation:<BR>\n" );
document.write( "    338x + 43((-126/198)x + (150/198)) = 303\r<BR>\n" );
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document.write( "3.  **Simplify and solve for x:**<BR>\n" );
document.write( "    338x - (5418/198)x + (6450/198) = 303<BR>\n" );
document.write( "    (66864/198)x - (5418/198)x = (60054/198) - (6450/198)<BR>\n" );
document.write( "    (61446/198)x = (53604/198)<BR>\n" );
document.write( "    x = 53604 / 61446<BR>\n" );
document.write( "    x ≈ 0.872\r<BR>\n" );
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document.write( "4.  **Substitute x back to find y:** Substitute the value of x into either of the original equations. Using the first equation is simpler.<BR>\n" );
document.write( "    126(0.872) + 198y = 150<BR>\n" );
document.write( "    110 + 198y = 150<BR>\n" );
document.write( "    198y = 40<BR>\n" );
document.write( "    y ≈ 0.202\r<BR>\n" );
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document.write( "**3. Elimination Method:**\r<BR>\n" );
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document.write( "1.  **Multiply equations to match coefficients:** We want to eliminate either x or y.  Let's eliminate y. Multiply the first equation by 43 and the second equation by 198:<BR>\n" );
document.write( "    *   (126x + 198y = 150) * 43  =>  5418x + 8514y = 6450<BR>\n" );
document.write( "    *   (338x + 43y = 303) * 198  =>  66864x + 8514y = 60054\r<BR>\n" );
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document.write( "2.  **Subtract the equations:** Subtract the first new equation from the second:<BR>\n" );
document.write( "    61446x = 53604\r<BR>\n" );
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document.write( "3.  **Solve for x:**<BR>\n" );
document.write( "    x ≈ 0.872\r<BR>\n" );
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document.write( "4.  **Substitute x back to find y:** Substitute the value of x into either of the original equations.<BR>\n" );
document.write( "    126(0.872) + 198y = 150<BR>\n" );
document.write( "    y ≈ 0.202\r<BR>\n" );
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document.write( "**4. Determinant Method (Cramer's Rule):**\r<BR>\n" );
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document.write( "1.  **Set up the coefficient matrix (D) and the matrices for x (Dx) and y (Dy):**\r<BR>\n" );
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document.write( "    D = | 126  198 |<BR>\n" );
document.write( "        | 338   43 |\r<BR>\n" );
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document.write( "    Dx = | 150  198 |<BR>\n" );
document.write( "         | 303   43 |\r<BR>\n" );
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document.write( "    Dy = | 126  150 |<BR>\n" );
document.write( "         | 338  303 |\r<BR>\n" );
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document.write( "2.  **Calculate the determinants:**<BR>\n" );
document.write( "    *   det(D) = (126 * 43) - (198 * 338) = -61446<BR>\n" );
document.write( "    *   det(Dx) = (150 * 43) - (198 * 303) = -53604<BR>\n" );
document.write( "    *   det(Dy) = (126 * 303) - (150 * 338) = -12300\r<BR>\n" );
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document.write( "3.  **Solve for x and y:**<BR>\n" );
document.write( "    x = det(Dx) / det(D) = -53604 / -61446 ≈ 0.872<BR>\n" );
document.write( "    y = det(Dy) / det(D) = -12300 / -61446 ≈ 0.200\r<BR>\n" );
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document.write( "**Solution:**\r<BR>\n" );
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document.write( "The solution using all methods converges to approximately:\r<BR>\n" );
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document.write( "*   x ≈ 0.872<BR>\n" );
document.write( "*   y ≈ 0.202<BR>\n" );
document.write( " </SPAN>\n" );
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