{"id":205,"date":"2017-08-23T09:59:03","date_gmt":"2017-08-23T09:59:03","guid":{"rendered":"http:\/\/americanboard.org\/Subjects\/mathematics\/?page_id=205"},"modified":"2017-09-18T19:28:17","modified_gmt":"2017-09-18T19:28:17","slug":"matrices-for-systems-of-equations","status":"publish","type":"page","link":"https:\/\/americanboard.org\/Subjects\/mathematics\/matrices-for-systems-of-equations\/","title":{"rendered":"Matrices for Systems of Equations"},"content":{"rendered":"
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\u2b05 Previous Lesson<\/a>\u00a0Workshop Index<\/a>\u00a0Next Lesson \u27a1<\/a><\/div>\n

<\/p>\n

Matrices for Systems of Equations<\/h1>\n

Objective<\/h4>\n

In this lesson, you will solve systems of linear equations using both the reduced row-echelon method and the\u00a0Gauss-Jordan elimination method.<\/p>\n

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W<\/span><\/em>ha<\/span>t\u00a0is reduced row-echelon form?<\/h3>\n

Reduced\u00a0row-echelon form<\/abbr> of a matrix is the form of a matrix in which:<\/p>\n

    \n
  1. any row with all zeros is at the bottom of the matrix.<\/li>\n
  2. any row that has an entry other than zero has one as the first non-zero entry.<\/li>\n
  3. any row that has one as the first non-zero entry has that\u00a0entry further to the right than the first non-zero entry of the\u00a0row above.<\/li>\n<\/ol>\n

    How do you change a matrix into reduced row-echelon form?<\/h3>\n

    There are a number of moves that can change a matrix into\u00a0reduced row-echelon form. But before we learn these moves, we will\u00a0first examine the connections between matrices and linear\u00a0equations.<\/p>\n

    For example, a school band decides to purchase xylophones and\u00a0zithers. If xylophones cost $30.00, and zithers cost $40.00, how\u00a0many of each should they buy in order to purchase exactly 254\u00a0instruments and spend their entire budget of $8,500?<\/p>\n

     <\/p>\n

    Two Perspectives on the Solution<\/p>\n

    Let <\/span>x\u00a0<\/span><\/em>= the number of xylophones. <\/span><\/p>\n

    Let <\/span>z\u00a0<\/span><\/em>= the number of zithers. <\/span><\/p>\n

    x<\/em> + z<\/em> = 254<\/p>\n

    30x<\/em> + 40z <\/em>= 8,500<\/p>\n

    Standard Elimination Method<\/p>\n

    Matrix Method<\/p>\n

    x<\/em> + z<\/em> = 254<\/p>\n

    30x<\/em> + 40z <\/em>= 8,500<\/p>\n

    <\/p>\n

    The above equation can also be represented<\/p>\n

    .<\/p>\n

    In order to solve the system using matrices, we use moves\u00a0(called row operations) similar to those on the left. The\u00a0following row operations are acceptable:<\/p>\n