document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=202458>Question 202458</A>: <I><SPAN STYLE=\"word-break: break-all;\"> Can anyone help me with these problems please?\r<BR>\n" );
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document.write( "1) Use the matrix method to solve:\r<BR>\n" );
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document.write( "5x + 3y = 3<BR>\n" );
document.write( "2x - 6y = 30\r<BR>\n" );
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document.write( "x = ____<BR>\n" );
document.write( "y = ____\r<BR>\n" );
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document.write( "2) Perform the following operations on matrices.\r<BR>\n" );
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document.write( "(4) + (8) = (____)<BR>\n" );
document.write( "(7)   (2)   (____)\r<BR>\n" );
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document.write( "3) Perform the following operations on matrices.\r<BR>\n" );
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document.write( "(1 8)(7 6) = (___ ___)<BR>\n" );
document.write( "(0 7)(7 4)   (___ ___)\r<BR>\n" );
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document.write( "Thank you in advance for all your help.<BR>\n" );
document.write( "~Sarah </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #152689 by </B><A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><B>jim_thompson5910(35256)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=jim_thompson5910><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=152689\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> # 1\r<BR>\n" );
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document.write( "First, convert the system\r<BR>\n" );
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document.write( "into the matrix equation\r<BR>\n" );
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document.write( "So what we now need is inverse matrix of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to isolate <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2Cx%2Cy%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C1%2Cx%2Cy%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " <TR><TD  ALIGN=MIDDLE>Solved by <A  rel=nofollow HREF=http://wiki.algebra.com/index.php/Tutoring_-_pluggable_solvers>pluggable</A> solver: <A  rel=nofollow HREF=inverse_of_2x2_matrix.solver>Finding the Inverse of a 2x2 Matrix</A></TD></TR>\n" );
document.write( "<TR><TD  ALIGN=LEFT><br>To find the inverse of the matrix <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, we can follow these steps:<br><br><h3>Step 1) Find the determinant</h3><br><br>The <a  rel=nofollow HREF=\"http://www.algebra.com/algebra/homework/Matrices-and-determiminant/determinant-of-2x2-matrix.solver\">determinant</a> of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=abs%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29=-36 ALIGN=MIDDLE  ALT=\"abs%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29=-36\"   DISABLED_style= \"max-width:90%; height:auto;\" >. So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-36 ALIGN=MIDDLE  ALT=\"d=-36\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><h3>Step 2) Swap the values</h3><br><br>Now switch the highlighted values <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Chighlight%285%29%2C3%2C2%2Chighlight%28-6%29%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Chighlight%285%29%2C3%2C2%2Chighlight%28-6%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Chighlight%28-6%29%2C3%2C2%2Chighlight%285%29%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Chighlight%28-6%29%2C3%2C2%2Chighlight%285%29%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><h3>Step 3) Change the sign</h3><br><br>Now change the sign of the highlighted values <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C-6%2Chighlight%283%29%2Chighlight%282%29%2C5%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C-6%2Chighlight%283%29%2Chighlight%282%29%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C-6%2Chighlight%28-3%29%2Chighlight%28-2%29%2C5%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C-6%2Chighlight%28-3%29%2Chighlight%28-2%29%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><h3>Step 4) Multiply by the inverse of the determinant</h3><br><br>Multiply by <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=1%2Fd ALIGN=MIDDLE  ALT=\"1%2Fd\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%2Fd%29%28matrix%282%2C2%2C-6%2C-3%2C-2%2C5%29%29 ALIGN=MIDDLE  ALT=\"%281%2Fd%29%28matrix%282%2C2%2C-6%2C-3%2C-2%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Plug in <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=d=-36 ALIGN=MIDDLE  ALT=\"d=-36\"   DISABLED_style= \"max-width:90%; height:auto;\" > to get   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28-1%2F36%29%28matrix%282%2C2%2C-6%2C-3%2C-2%2C5%29%29 ALIGN=MIDDLE  ALT=\"%28-1%2F36%29%28matrix%282%2C2%2C-6%2C-3%2C-2%2C5%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><h3>Step 5) Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F36 ALIGN=MIDDLE  ALT=\"-1%2F36\"   DISABLED_style= \"max-width:90%; height:auto;\" > by every element in the matrix (simplify and reduce if possible)</h3><br><br>Multiply <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-1%2F36 ALIGN=MIDDLE  ALT=\"-1%2F36\"   DISABLED_style= \"max-width:90%; height:auto;\" > by EVERY element to get   <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C%28-1%2F36%29%28-6%29%2C%28-1%2F36%29%28-3%29%2C%28-1%2F36%29%28-2%29%2C%28-1%2F36%29%285%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Multiply to get <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C-6%2F-36%2C-3%2F-36%2C-2%2F-36%2C5%2F-36%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C-6%2F-36%2C-3%2F-36%2C-2%2F-36%2C5%2F-36%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>Reduce each element: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br><br>=================================================================<br><br><br>Answer:<br><br>So the inverse of <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br><br>This means that if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29 ALIGN=MIDDLE  ALT=\"A=%28matrix%282%2C2%2C5%2C3%2C2%2C-6%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" > then <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=A%5E%28-1%29=%28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29 ALIGN=MIDDLE  ALT=\"A%5E%28-1%29=%28matrix%282%2C2%2C1%2F6%2C1%2F12%2C1%2F18%2C-5%2F36%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" ><br></TD></TR>\n" );
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document.write( "Now left multiply both sides by the inverse matrix to get\r<BR>\n" );
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document.write( "So this means that <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x=3 ALIGN=MIDDLE  ALT=\"x=3\"   DISABLED_style= \"max-width:90%; height:auto;\" > and <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=y=-4 ALIGN=MIDDLE  ALT=\"y=-4\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "Note: let me know if you need help with matrix multiplication.\r<BR>\n" );
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document.write( "# 2\r<BR>\n" );
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document.write( "Simply add the corresponding components:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C4%2C7%29%29%2B%28matrix%282%2C1%2C8%2C2%29%29=%28matrix%282%2C1%2C4%2B8%2C7%2B2%29%29=%28matrix%282%2C1%2C12%2C9%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "So <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C1%2C4%2C7%29%29%2B%28matrix%282%2C1%2C8%2C2%29%29=%28matrix%282%2C1%2C12%2C9%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "# 3\r<BR>\n" );
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document.write( "Since the first matrix is a 2 by 2 matrix and the second matrix is a 2 by 2 matrix, this means that the resulting matrix will be a 2 by  2 matrix.\r<BR>\n" );
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document.write( "So the final resulting matrix will look like:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Cx%2Cx%2Cx%2Cx%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2Cx%2Cx%2Cx%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "note: the \"x\"s are just placeholders for now\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">1st</font> row of the first matrix by the <font size=\"4\" color=\"green\">1st</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "<font size=\"4\" color=\"red\">1st</font> row, <font size=\"4\" color=\"green\">1st</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2Chighlight%281%29%2Chighlight%288%29%2C0%2C7%29%29%28matrix%282%2C2%2Chighlight%287%29%2C6%2Chighlight%287%29%2C4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%287%29%2B%288%29%2A%287%29=7%2B56=63 ALIGN=MIDDLE  ALT=\"%281%29%2A%287%29%2B%288%29%2A%287%29=7%2B56=63\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the <font size=\"4\" color=\"red\">1st</font> row, <font size=\"4\" color=\"green\">1st</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=63 ALIGN=MIDDLE  ALT=\"63\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">1st</font> row of the first matrix by the <font size=\"4\" color=\"green\">2nd</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%281%29%2A%286%29%2B%288%29%2A%284%29=6%2B32=38 ALIGN=MIDDLE  ALT=\"%281%29%2A%286%29%2B%288%29%2A%284%29=6%2B32=38\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the <font size=\"4\" color=\"red\">1st</font> row, <font size=\"4\" color=\"green\">2nd</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=38 ALIGN=MIDDLE  ALT=\"38\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "================================================================================\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">2nd</font> row of the first matrix by the <font size=\"4\" color=\"green\">1st</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "<font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">1st</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2C8%2Chighlight%280%29%2Chighlight%287%29%29%29%28matrix%282%2C2%2Chighlight%287%29%2C6%2Chighlight%287%29%2C4%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%2A%287%29%2B%287%29%2A%287%29=0%2B49=49 ALIGN=MIDDLE  ALT=\"%280%29%2A%287%29%2B%287%29%2A%287%29=0%2B49=49\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the <font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">1st</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=49 ALIGN=MIDDLE  ALT=\"49\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C63%2C38%2C49%2Cx%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C63%2C38%2C49%2Cx%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "--------------------------------------------------\r<BR>\n" );
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document.write( "Multiply the corresponding entries from the <font size=\"4\" color=\"red\">2nd</font> row of the first matrix by the <font size=\"4\" color=\"green\">2nd</font> column of the second matrix. After multiplying, add the values:\r<BR>\n" );
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document.write( "<font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">2nd</font> column: <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C1%2C8%2Chighlight%280%29%2Chighlight%287%29%29%29%28matrix%282%2C2%2C7%2Chighlight%286%29%2C7%2Chighlight%284%29%29%29 ALIGN=MIDDLE   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%280%29%2A%286%29%2B%287%29%2A%284%29=0%2B28=28 ALIGN=MIDDLE  ALT=\"%280%29%2A%286%29%2B%287%29%2A%284%29=0%2B28=28\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( " So the element in the <font size=\"4\" color=\"red\">2nd</font> row, <font size=\"4\" color=\"green\">2nd</font> column of the resulting matrix is <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=28 ALIGN=MIDDLE  ALT=\"28\"   DISABLED_style= \"max-width:90%; height:auto;\" >. Now let's update the matrix:\r<BR>\n" );
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document.write( "==============================================================================\r<BR>\n" );
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document.write( "Answer:\r<BR>\n" );
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document.write( "So the solution is  <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=%28matrix%282%2C2%2C63%2C38%2C49%2C28%29%29 ALIGN=MIDDLE  ALT=\"%28matrix%282%2C2%2C63%2C38%2C49%2C28%29%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
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document.write( "In other words,\r<BR>\n" );
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