{"id":467,"date":"2016-04-21T22:43:38","date_gmt":"2016-04-21T22:43:38","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/introstats1xmaster\/?post_type=chapter&p=467"},"modified":"2019-01-01T18:08:54","modified_gmt":"2019-01-01T18:08:54","slug":"linear-equations","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/suny-fmcc-introstats1\/chapter\/linear-equations\/","title":{"raw":"Linear Equations","rendered":"Linear Equations"},"content":{"raw":"Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:\r\n
\r\n
y<\/span>=<\/span>a<\/span>+<\/span>bx<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\r\n<\/div>\r\n
<\/div>\r\nwhere a<\/em> and b<\/em> are constant numbers.\r\n\r\nThe variable x<\/em> is the independent variable, and y<\/em> is the dependent variable.<\/strong> Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.\r\n
The following examples are linear equations.\r\n
\r\n
y<\/span>=<\/span>3<\/span>+<\/span>2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\r\n<\/div>\r\n
\r\n
y<\/span>=<\/span>\u20130.01<\/span>+<\/span>1.2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n
\r\n
\r\n
\r\n

try it<\/h3>\r\n
\r\n
\r\n
\r\n\r\nIs the following an example of a linear equation?\r\n

y<\/em> = \u20130.125 \u2013 3.5x<\/em><\/p>\r\n\r\n<\/div>\r\n

<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n

The graph of a linear equation of the form y<\/em> = a<\/em> + bx<\/em> is a straight line<\/strong>. Any line that is not vertical can be described by this equation.<\/p>\r\n\r\n

\r\n

Graph the equation y<\/em> = \u20131 + 2x<\/em>.<\/p>\r\n\r\n

\"Graph<\/span><\/figure>\r\n<\/section><\/div>\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n<\/header><\/div>\r\n
\r\n
\r\n
\r\n

try it<\/h3>\r\n
\r\n
\r\n
\r\n
    \r\n \t
  1. Is the following an example of a linear equation? Why or why not?<\/li>\r\n<\/ol>\r\n\"This<\/span>\r\n\r\nAaron's Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n
    \r\n
    \r\n
    \r\n

    Find the equation that expresses the total cost<\/strong> in terms of the number of hours<\/strong>required to complete the job.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n \r\n\r\n<\/div>\r\n<\/header><\/div>\r\n

    \r\n
    \r\n
    \r\n

    try it<\/h3>\r\n
    \r\n
    \r\n
    \r\n\r\nEmma\u2019s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class. Find the equation that expresses the total cost in terms of the number of students in a class.\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n
    \r\n

    Slope and Y<\/em>-Intercept of a Linear Equation<\/h1>\r\n

    For the linear equation y<\/em> = a<\/em> + bx<\/em>, b<\/em> = slope and a<\/em> = y<\/em>-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the y<\/em>-intercept is the y<\/em> coordinate of the point (0, a<\/em>) where the line crosses the y<\/em>-axis.<\/p>\r\n\r\n

    \"Three<\/span>
    Three possible graphs of y<\/em> = a<\/em> + bx<\/em>. (a) If b<\/em> > 0, the line slopes upward to the right. (b) If b<\/em> = 0, the line is horizontal. (c) If b<\/em> < 0, the line slopes downward to the right.<\/figcaption><\/figure>\r\n
    Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y<\/em> = 25 + 15x<\/em>.\r\n
    \r\n
    \r\n

    What are the independent and dependent variables? What is the y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n

    \r\n
    \r\n
    \r\n

    try it<\/h3>\r\n
    \r\n
    \r\n
    \r\n

    Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is y<\/em> = 25 + 20x<\/em>.<\/p>\r\n

    What are the independent and dependent variables? What is the y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/header><\/div>\r\n<\/section>

    \r\n

    References<\/h1>\r\n

    Data from the Centers for Disease Control and Prevention.<\/p>\r\n

    Data from the National Center for HIV, STD, and TB Prevention.<\/p>\r\n\r\n<\/section>

    \r\n

    Concept Review<\/h1>\r\n

    The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form y = mx + b<\/em><\/strong>, where m<\/em><\/strong> and b<\/em><\/strong> are constants, x<\/em><\/strong> is the independent variable, y<\/em><\/strong> is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx<\/em><\/strong>, where a<\/em><\/strong>and b<\/em><\/strong> are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation y = a + bx<\/em>, the constant b<\/em> that multiplies the x<\/em><\/strong> variable (b<\/em> is called a coefficient) is called as the slope<\/strong>. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx<\/em>, the constant a is called as the y<\/em>-intercept. Graphically, the y<\/em>-intercept is the y<\/em> coordinate of the point where the graph of the line crosses the y<\/em> axis. At this point x<\/em> = 0.<\/p>\r\nThe slope of a line<\/strong> is a value that describes the rate of change between the independent and dependent variables. The slope<\/strong> tells us how the dependent variable (y<\/em>) changes for every one unit increase in the independent (x<\/em>) variable, on average. The y<\/em>-intercept<\/strong> is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.\r\n\r\n<\/section>

    \r\n

    Formula Review<\/h1>\r\ny<\/em> = a<\/em> + bx<\/em> where a<\/em> is the y<\/em>-intercept and b<\/em> is the slope. The variable x<\/em> is the independent variable andy<\/em> is the dependent variable.\r\n\r\n<\/section>
    <\/section>
    <\/section>","rendered":"

    Linear regression for two variables is based on a linear equation with one independent variable. The equation has the form:<\/p>\n

    \n
    y<\/span>=<\/span>a<\/span>+<\/span>bx<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\n<\/div>\n
    <\/div>\n

    where a<\/em> and b<\/em> are constant numbers.<\/p>\n

    The variable x<\/em> is the independent variable, and y<\/em> is the dependent variable.<\/strong> Typically, you choose a value to substitute for the independent variable and then solve for the dependent variable.<\/p>\n

    \n
    The following examples are linear equations.<\/p>\n
    \n
    y<\/span>=<\/span>3<\/span>+<\/span>2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\n<\/div>\n
    \n
    y<\/span>=<\/span>\u20130.01<\/span>+<\/span>1.2x<\/span><\/span><\/span><\/span><\/span><\/span><\/div>\n<\/div>\n<\/section>\n<\/div>\n
    \n
    \n
    \n
    \n

    try it<\/h3>\n
    \n
    \n
    \n
    \n
    \n

    Is the following an example of a linear equation?<\/p>\n

    y<\/em> = \u20130.125 \u2013 3.5x<\/em><\/p>\n<\/div>\n

    <\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n

    The graph of a linear equation of the form y<\/em> = a<\/em> + bx<\/em> is a straight line<\/strong>. Any line that is not vertical can be described by this equation.<\/p>\n

    \n
    \n

    Graph the equation y<\/em> = \u20131 + 2x<\/em>.<\/p>\n

    \"Graph<\/span><\/figure>\n<\/section>\n<\/div>\n<\/div>\n

     <\/p>\n<\/div>\n<\/header>\n<\/div>\n

    \n
    \n
    \n
    \n

    try it<\/h3>\n
    \n
    \n
    \n
    \n
    \n
      \n
    1. Is the following an example of a linear equation? Why or why not?<\/li>\n<\/ol>\n

      \"This<\/span><\/p>\n

      Aaron’s Word Processing Service (AWPS) does word processing. The rate for services is $32 per hour \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0 \u00a0plus a $31.50 one-time charge. The total cost to a customer depends on the number of hours it takes to complete the job.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n

      \n
      \n
      \n
      \n
      \n

      Find the equation that expresses the total cost<\/strong> in terms of the number of hours<\/strong>required to complete the job.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n

       <\/p>\n<\/div>\n<\/header>\n<\/div>\n

      \n
      \n
      \n
      \n

      try it<\/h3>\n
      \n
      \n
      \n
      \n

      Emma\u2019s Extreme Sports hires hang-gliding instructors and pays them a fee of $50 per class as well as $20 per student in the class. The total cost Emma pays depends on the number of students in a class. Find the equation that expresses the total cost in terms of the number of students in a class.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n

      \n

      Slope and Y<\/em>-Intercept of a Linear Equation<\/h1>\n

      For the linear equation y<\/em> = a<\/em> + bx<\/em>, b<\/em> = slope and a<\/em> = y<\/em>-intercept. From algebra recall that the slope is a number that describes the steepness of a line, and the y<\/em>-intercept is the y<\/em> coordinate of the point (0, a<\/em>) where the line crosses the y<\/em>-axis.<\/p>\n

      \"Three<\/span>
      Three possible graphs of y<\/em> = a<\/em> + bx<\/em>. (a) If b<\/em> > 0, the line slopes upward to the right. (b) If b<\/em> = 0, the line is horizontal. (c) If b<\/em> < 0, the line slopes downward to the right.<\/figcaption><\/figure>\n
      \n
      Svetlana tutors to make extra money for college. For each tutoring session, she charges a one-time fee of $25 plus $15 per hour of tutoring. A linear equation that expresses the total amount of money Svetlana earns for each session she tutors is y<\/em> = 25 + 15x<\/em>.<\/p>\n
      \n
      \n
      \n

      What are the independent and dependent variables? What is the y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n

      \n
      \n
      \n
      \n

      try it<\/h3>\n
      \n
      \n
      \n
      \n

      Ethan repairs household appliances like dishwashers and refrigerators. For each visit, he charges $25 plus $20 per hour of work. A linear equation that expresses the total amount of money Ethan earns per visit is y<\/em> = 25 + 20x<\/em>.<\/p>\n

      What are the independent and dependent variables? What is the y<\/em>-intercept and what is the slope? Interpret them using complete sentences.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/header>\n<\/div>\n<\/section>\n

      \n

      References<\/h1>\n

      Data from the Centers for Disease Control and Prevention.<\/p>\n

      Data from the National Center for HIV, STD, and TB Prevention.<\/p>\n<\/section>\n

      \n

      Concept Review<\/h1>\n

      The most basic type of association is a linear association. This type of relationship can be defined algebraically by the equations used, numerically with actual or predicted data values, or graphically from a plotted curve. (Lines are classified as straight curves.) Algebraically, a linear equation typically takes the form y = mx + b<\/em><\/strong>, where m<\/em><\/strong> and b<\/em><\/strong> are constants, x<\/em><\/strong> is the independent variable, y<\/em><\/strong> is the dependent variable. In a statistical context, a linear equation is written in the form y = a + bx<\/em><\/strong>, where a<\/em><\/strong>and b<\/em><\/strong> are the constants. This form is used to help readers distinguish the statistical context from the algebraic context. In the equation y = a + bx<\/em>, the constant b<\/em> that multiplies the x<\/em><\/strong> variable (b<\/em> is called a coefficient) is called as the slope<\/strong>. The slope describes the rate of change between the independent and dependent variables; in other words, the rate of change describes the change that occurs in the dependent variable as the independent variable is changed. In the equation y = a + bx<\/em>, the constant a is called as the y<\/em>-intercept. Graphically, the y<\/em>-intercept is the y<\/em> coordinate of the point where the graph of the line crosses the y<\/em> axis. At this point x<\/em> = 0.<\/p>\n

      The slope of a line<\/strong> is a value that describes the rate of change between the independent and dependent variables. The slope<\/strong> tells us how the dependent variable (y<\/em>) changes for every one unit increase in the independent (x<\/em>) variable, on average. The y<\/em>-intercept<\/strong> is used to describe the dependent variable when the independent variable equals zero. Graphically, the slope is represented by three line types in elementary statistics.<\/p>\n<\/section>\n

      \n

      Formula Review<\/h1>\n

      y<\/em> = a<\/em> + bx<\/em> where a<\/em> is the y<\/em>-intercept and b<\/em> is the slope. The variable x<\/em> is the independent variable andy<\/em> is the dependent variable.<\/p>\n<\/section>\n

      <\/section>\n
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      Candela Citations<\/h3>\n\t\t\t\t\t
      \n\t\t\t\t\t\t
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      CC licensed content, Shared previously<\/div>