{"id":7092,"date":"2014-12-16T18:55:40","date_gmt":"2014-12-16T18:55:40","guid":{"rendered":"http:\/\/www.oracletutoring.ca\/blog\/?p=7092"},"modified":"2018-02-15T19:07:49","modified_gmt":"2018-02-15T19:07:49","slug":"math-solving-systems-of-equations-cramers-rule","status":"publish","type":"post","link":"https:\/\/www.oracletutoring.ca\/blog\/math-solving-systems-of-equations-cramers-rule\/","title":{"rendered":"Math:  Solving systems of equations:  Cramer&#8217;s Rule"},"content":{"rendered":"<h1>Tutoring high school math, I don&#8217;t see this used. \u00a0However, I did see it at university. \u00a0The tutor introduces Cramer&#8217;s Rule.<\/h1>\n<p>Consider the following problem, common in high school math:<\/p>\n<p><strong>Solve the system.<\/strong><\/p>\n<p>2x-4y=20<br \/>\n3x+5y=-3<\/p>\n<p>If you know determinants (having read my articles <a href=\"?p=6543\">here<\/a> and <a href=\"?p=6583\">here<\/a>), then you have the option to use Cramer&#8217;s Rule.<\/p>\n<p>Cramer&#8217;s Rule sees the problem as follows:<\/p>\n<p>The original system, in matrix form, is called the augmented matrix &#8211; so-called because it contains the constant numbers in the third column:<\/p>\n<p><img decoding=\"async\" src=\"\/..\/images\/cramer_matrix_feb15_2018.png\" \/><\/p>\n<p>The matrix with just the x and y columns is called &#8220;the 2&#215;2 matrix&#8221;:<\/p>\n<p><img decoding=\"async\" src=\"\/..\/images\/cramer_2by2_matrix_feb15_2018.png\" \/><\/p>\n<p>With Cramer&#8217;s Rule, we evaluate determinants of various matrices derived from the augmented matrix.  The solution to each variable begins in fraction form.  Its numerator is a 2&#215;2 determinant in which the third column from the augmented matrix is substituted for the column of the variable being found.  The denominator is the determinant of the 2&#215;2 matrix.  Let&#8217;s proceed:<\/p>\n<p>Solving the above system for x, Cramer&#8217;s Rule gives<\/p>\n<p><img decoding=\"async\" src=\"\/..\/images\/cramer_xsol_feb15_2018.png\" \/><\/p>\n<p>Notice that, in the numerator, the third column from the augmented matrix is substituted for the $x$ column, while the $y$ column stays the same.<\/p>\n<p>Then, Cramer&#8217;s Rule gives, for y:<\/p>\n<p><img decoding=\"async\" src=\"\/..\/images\/cramer_ysol_feb15_2018.png\" \/><\/p>\n<p>We evaluate the determinants in each case, then simplify.  First for x:<\/p>\n<p>x=(100-12)\/(10+12)=88\/22=4<\/p>\n<p>We do the same for y:<\/p>\n<p>y=(-6-60)\/(10+12)=-66\/22=-3<\/p>\n<p>Apparently, x=4,y=-3.  Let&#8217;s sub the values back into our original equations just to make sure:<\/p>\n<p>2(4)-4(-3)=20 correct.<\/p>\n<p>3(4)+5(-3)=-3 correct.<\/p>\n<p>Since, in each of our original equations, the left side equals the right side, we have confirmation that our solution x=4,y=-3 is correct.<\/p>\n<p>Cramer&#8217;s Rule can be very handy, especially for 2-variable systems.<\/p>\n<p>I&#8217;ll be saying more about Cramer&#8217;s Rule and systems of equations in future posts:)<\/p>\n<p><em>Source<\/em>:<\/p>\n<p>Johnson\/Riess\/Arnold.  <em>Introduction to Linear Algebra<\/em>, 2nd edition.  Don Mills: Addison-Wesley, 1989.<\/p>\n<p>Jack of <a href=\"https:\/\/www.oracletutoring.ca\">Oracle Tutoring by Jack and Diane,<\/a> Campbell River, BC.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>Tutoring high school math, I don&#8217;t see this used. \u00a0However, I did see it at university. \u00a0The tutor introduces Cramer&#8217;s Rule. Consider the following problem, common in high school math: Solve the system. 2x-4y=20 3x+5y=-3 If you know determinants (having &hellip;<\/p>\n<p class=\"read-more\"> <a class=\"more-link\" href=\"https:\/\/www.oracletutoring.ca\/blog\/math-solving-systems-of-equations-cramers-rule\/\"> <span class=\"screen-reader-text\">Math:  Solving systems of equations:  Cramer&#8217;s Rule<\/span> Read More &raquo;<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[302,3],"tags":[362,74],"class_list":["post-7092","post","type-post","status-publish","format-standard","hentry","category-linear-algebra","category-math","tag-cramers-rule","tag-systems-of-equations"],"_links":{"self":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/7092","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/comments?post=7092"}],"version-history":[{"count":57,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/7092\/revisions"}],"predecessor-version":[{"id":29843,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/posts\/7092\/revisions\/29843"}],"wp:attachment":[{"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/media?parent=7092"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/categories?post=7092"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.oracletutoring.ca\/blog\/wp-json\/wp\/v2\/tags?post=7092"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}