{"id":104,"date":"2021-01-12T22:19:50","date_gmt":"2021-01-12T22:19:50","guid":{"rendered":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/linear-equations\/"},"modified":"2023-04-19T20:25:57","modified_gmt":"2023-04-19T20:25:57","slug":"linear-equations","status":"publish","type":"chapter","link":"https:\/\/textbooks.jaykesler.net\/introstats\/chapter\/linear-equations\/","title":{"rendered":"Linear Equations"},"content":{"raw":"\r\n[latexpage]\r\n<\/span>\r\n
\r\n

Linear regression for two variables is based on a linear equation with one independent\r\nvariable. The equation has the form:<\/p>\r\n$$y=a+bx$$\r\n\r\nwhere a<\/em> and b<\/em> are constant numbers.\r\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>\r\n\r\n

\r\n

Example <\/span>12.1<\/span><\/h3>\r\n<\/header>
\r\n
\r\n

The following examples are linear equations.<\/p>\r\n$$y=3+2x$$\r\n\r\n$$y=-0.01+1.2x$$\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n

\r\n

Try It <\/span>12.1<\/span><\/h3>\r\n<\/header>
\r\n
<\/header>
\r\n
\r\n
\r\n

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

\r\n

Example <\/span>12.2<\/span><\/h3>\r\n<\/header>
\r\n
\r\n

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

\r\n
\r\n\"Graph\r\n<\/span><\/figure>\r\n
Figure <\/span>12.2<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n
\r\n

Try It <\/span>12.2<\/span><\/h3>\r\n<\/header>
\r\n
<\/header>
\r\n
\r\n
\r\n

Is the following an example of a linear equation? Why or why not?<\/p>\r\n\r\n

\r\n
\r\n\"This\r\n<\/span><\/figure>\r\n
Figure <\/span>12.3<\/span><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n
\r\n

Example <\/span>12.3<\/span><\/h3>\r\n<\/header>
\r\n
\r\n

Aaron's Word Processing Service (AWPS) does word processing. The rate for services is \\$32 per hour plus 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>\r\n\r\n

<\/header>
\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<\/div>\r\n

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

Solution <\/span>12.3<\/span><\/h4>\r\n
\r\n

Let x<\/em> = the number of hours it takes to get the job done.\r\n\r\n<\/span>Let y<\/em> = the total cost to the customer.<\/p>\r\n

The \\$31.50 is a fixed cost. If it takes x<\/em> hours to complete the job, then (32)(x<\/em>) is the cost of the word processing only. The total cost is: y<\/em> = 31.50 + 32x<\/em><\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n

\r\n

Try It <\/span>12.3<\/span><\/h3>\r\n<\/header>
\r\n
<\/header>
\r\n
\r\n
\r\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>\r\n\r\n<\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n

\r\n

Slope and Y<\/em>-Intercept of a Linear Equation<\/h3>\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

\r\n
\"Three<\/span><\/figure>\r\n
Figure <\/span>12.4<\/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.<\/span><\/div>\r\n<\/div>\r\n
\r\n

Example <\/span>12.4<\/span><\/h3>\r\n<\/header>
\r\n
\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>.<\/p>\r\n\r\n

<\/header>
\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<\/div>\r\n

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

Solution <\/span>12.4<\/span><\/h4>\r\n
\r\n

The independent variable (x<\/em>) is the number of hours Svetlana tutors each session. The dependent variable (y<\/em>) is the amount, in dollars, Svetlana earns for each session.<\/p>\r\n

The y<\/em>-intercept is 25 (a<\/em> = 25). At the start of the tutoring session, Svetlana charges a one-time fee of \\$25 (this is when x<\/em> = 0). The slope is 15 (b<\/em> = 15). For each session, Svetlana earns \\$15 for each hour she tutors.<\/p>\r\n\r\n<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/section><\/div>\r\n

\r\n

Try It <\/span>12.4<\/span><\/h3>\r\n<\/header>
\r\n
<\/header>
\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<\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>\r\n<\/section><\/div>","rendered":"


\n[latexpage]
\n<\/span><\/p>\n

\n

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

$$y=a+bx$$<\/p>\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
\n

Example <\/span>12.1<\/span><\/h3>\n<\/header>\n
\n
\n

The following examples are linear equations.<\/p>\n

$$y=3+2x$$<\/p>\n

$$y=-0.01+1.2x$$<\/p>\n<\/div>\n<\/section>\n<\/div>\n

\n
\n

Try It <\/span>12.1<\/span><\/h3>\n<\/header>\n
\n
\n
<\/header>\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 some linear equation.<\/p>\n

\n
\n

Example <\/span>12.2<\/span><\/h3>\n<\/header>\n
\n
\n

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

\n

\n\"Graph
\n<\/span><\/figure>\n
Figure <\/span>12.2<\/span><\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n
\n
\n

Try It <\/span>12.2<\/span><\/h3>\n<\/header>\n
\n
\n
<\/header>\n
\n
\n
\n

Is the following an example of a linear equation? Why or why not?<\/p>\n

\n

\n\"This
\n<\/span><\/figure>\n
Figure <\/span>12.3<\/span><\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n
\n
\n

Example <\/span>12.3<\/span><\/h3>\n<\/header>\n
\n
\n

Aaron’s Word Processing Service (AWPS) does word processing. The rate for services is \\$32 per hour plus 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

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

\n
<\/div>\n
\n

Solution <\/span>12.3<\/span><\/h4>\n
\n

Let x<\/em> = the number of hours it takes to get the job done.
\n
\n<\/span>Let y<\/em> = the total cost to the customer.<\/p>\n

The \\$31.50 is a fixed cost. If it takes x<\/em> hours to complete the job, then (32)(x<\/em>) is the cost of the word processing only. The total cost is: y<\/em> = 31.50 + 32x<\/em><\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n

\n
\n

Try It <\/span>12.3<\/span><\/h3>\n<\/header>\n
\n
\n
<\/header>\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<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n

\n

Slope and Y<\/em>-Intercept of a Linear Equation<\/h3>\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

\n
\"Three<\/span><\/figure>\n
Figure <\/span>12.4<\/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.<\/span><\/div>\n<\/div>\n
\n
\n

Example <\/span>12.4<\/span><\/h3>\n<\/header>\n
\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
<\/header>\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<\/div>\n

\n
<\/div>\n
\n

Solution <\/span>12.4<\/span><\/h4>\n
\n

The independent variable (x<\/em>) is the number of hours Svetlana tutors each session. The dependent variable (y<\/em>) is the amount, in dollars, Svetlana earns for each session.<\/p>\n

The y<\/em>-intercept is 25 (a<\/em> = 25). At the start of the tutoring session, Svetlana charges a one-time fee of \\$25 (this is when x<\/em> = 0). The slope is 15 (b<\/em> = 15). For each session, Svetlana earns \\$15 for each hour she tutors.<\/p>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/div>\n<\/section>\n<\/div>\n

\n
\n

Try It <\/span>12.4<\/span><\/h3>\n<\/header>\n
\n
\n
<\/header>\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<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n<\/section>\n<\/div>\n","protected":false},"author":1,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-104","chapter","type-chapter","status-publish","hentry"],"part":103,"_links":{"self":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/104","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/users\/1"}],"version-history":[{"count":5,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/104\/revisions"}],"predecessor-version":[{"id":583,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/104\/revisions\/583"}],"part":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/parts\/103"}],"metadata":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapters\/104\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/media?parent=104"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/pressbooks\/v2\/chapter-type?post=104"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/contributor?post=104"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/textbooks.jaykesler.net\/introstats\/wp-json\/wp\/v2\/license?post=104"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}