document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1205173>Question 1205173</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Consider the following system of linear equations<BR>\n" );
document.write( "𝑥 + 3𝑦 + 2𝑧 − 𝑤 = −1<BR>\n" );
document.write( "−3𝑥 − 7𝑦 + (𝑝 − 6)𝑧 + 2𝑤 = 1<BR>\n" );
document.write( "2𝑥 + 𝑝^2𝑧 + 𝑝𝑤 = 𝑞^2<BR>\n" );
document.write( "where 𝑝 and 𝑞 are real numbers.<BR>\n" );
document.write( "Using Gaussian elimination, determine all possible values of 𝑝 and 𝑞 such that the system has infinitely many solutions with two free variables and solve the systems </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #847979 by </B><A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><B>ElectricPavlov(122)</B></A><img src=\"/images/stars/stars-09.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=ElectricPavlov><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/images/nopicture.jpg><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=847979\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> **1. Set up the Augmented Matrix**\r<BR>\n" );
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document.write( "The given system of linear equations can be represented by the following augmented matrix:\r<BR>\n" );
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document.write( "```<BR>\n" );
document.write( "[ 1   3   2  -1 | -1 ]<BR>\n" );
document.write( "[-3  -7  p-6   2 |  1 ]<BR>\n" );
document.write( "[ 2   0  p^2   p | q^2 ]<BR>\n" );
document.write( "```\r<BR>\n" );
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document.write( "**2. Perform Row Operations**\r<BR>\n" );
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document.write( "* **R2 = R2 + 3R1:** <BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   [ 1   3   2  -1 | -1 ]<BR>\n" );
document.write( "   [ 0   2  p     -1 | -2 ]<BR>\n" );
document.write( "   [ 2   0  p^2   p | q^2 ]<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "* **R3 = R3 - 2R1:**<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   [ 1   3   2  -1 | -1 ]<BR>\n" );
document.write( "   [ 0   2   p     -1 | -2 ]<BR>\n" );
document.write( "   [ 0  -6  p^2-4  p+2 | q^2+2 ]<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "* **R3 = R3 + 3R2:**<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   [ 1   3   2  -1 | -1 ]<BR>\n" );
document.write( "   [ 0   2   p     -1 | -2 ]<BR>\n" );
document.write( "   [ 0   0  p^2+3p-4  p-1 | q^2-4 ]<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "**3. Analyze for Infinitely Many Solutions**\r<BR>\n" );
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document.write( "For the system to have infinitely many solutions with two free variables, the last row of the augmented matrix must be a row of zeros. This implies:\r<BR>\n" );
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document.write( "* p^2 + 3p - 4 = 0 <BR>\n" );
document.write( "* p - 1 = 0 <BR>\n" );
document.write( "* q^2 - 4 = 0\r<BR>\n" );
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document.write( "**4. Solve for p and q**\r<BR>\n" );
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document.write( "* **p^2 + 3p - 4 = 0** <BR>\n" );
document.write( "   * (p + 4)(p - 1) = 0 <BR>\n" );
document.write( "   * p = -4 or p = 1\r<BR>\n" );
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document.write( "* **p - 1 = 0** <BR>\n" );
document.write( "   * p = 1\r<BR>\n" );
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document.write( "* **q^2 - 4 = 0** <BR>\n" );
document.write( "   * q = ±2\r<BR>\n" );
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document.write( "Since we need all conditions to be satisfied simultaneously, the only possible values for p and q are:\r<BR>\n" );
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document.write( "* **p = 1** <BR>\n" );
document.write( "* **q = 2 or q = -2**\r<BR>\n" );
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document.write( "**5. Solve the System for p = 1, q = 2 (or q = -2)**\r<BR>\n" );
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document.write( "* Substitute p = 1 into the row-echelon form of the augmented matrix:<BR>\n" );
document.write( "   ```<BR>\n" );
document.write( "   [ 1   3   2  -1 | -1 ]<BR>\n" );
document.write( "   [ 0   2   1     -1 | -2 ]<BR>\n" );
document.write( "   [ 0   0   0     0 | 0 ]<BR>\n" );
document.write( "   ```\r<BR>\n" );
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document.write( "* Let z = s and w = t (where s and t are free variables).<BR>\n" );
document.write( "* From the second row: 2y + s - t = -2 <BR>\n" );
document.write( "   => y = (-s + t - 2)/2\r<BR>\n" );
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document.write( "* From the first row: x + 3y + 2z - w = -1<BR>\n" );
document.write( "   => x + 3[(-s + t - 2)/2] + 2s - t = -1 <BR>\n" );
document.write( "   => x = (s - t - 1)/2\r<BR>\n" );
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document.write( "**Therefore, the general solution for p = 1 and q = ±2 is:**\r<BR>\n" );
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document.write( "* x = (s - t - 1)/2<BR>\n" );
document.write( "* y = (-s + t - 2)/2<BR>\n" );
document.write( "* z = s<BR>\n" );
document.write( "* w = t\r<BR>\n" );
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document.write( "where s and t are arbitrary real numbers.\r<BR>\n" );
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document.write( "**In summary:**\r<BR>\n" );
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document.write( "* The system of linear equations has infinitely many solutions with two free variables only when p = 1 and q = 2 or q = -2. <BR>\n" );
document.write( "* The general solution for these values of p and q is given above.<BR>\n" );
document.write( " </SPAN>\n" );
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