{"id":3089,"date":"2019-10-23T14:05:21","date_gmt":"2019-10-23T14:05:21","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/mathforlibscoreq\/?post_type=chapter&#038;p=3089"},"modified":"2021-02-05T23:54:49","modified_gmt":"2021-02-05T23:54:49","slug":"summary-review-topics-4","status":"web-only","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/chapter\/summary-review-topics-4\/","title":{"raw":"Summary: Review Topics","rendered":"Summary: Review Topics"},"content":{"raw":"<h2>Key Concepts<\/h2>\r\n<ul>\r\n \t<li><strong>Sign Patterns of the Quadrants<\/strong>\r\n<table id=\"eip-id1170324021306\" class=\"unnumbered unstyled\" summary=\"...\">\r\n<thead>\r\n<tr>\r\n<th>Quadrant I<\/th>\r\n<th>Quadrant II<\/th>\r\n<th>Quadrant III<\/th>\r\n<th>Quadrant IV<\/th>\r\n<\/tr>\r\n<\/thead>\r\n<tbody>\r\n<tr>\r\n<td>[latex](x,y)[\/latex]<\/td>\r\n<td>[latex](x,y)[\/latex]<\/td>\r\n<td>[latex](x,y)[\/latex]<\/td>\r\n<td>[latex](x,y)[\/latex]<\/td>\r\n<\/tr>\r\n<tr>\r\n<td>[latex](+,+)[\/latex]<\/td>\r\n<td>[latex](\u2212,+)[\/latex]<\/td>\r\n<td>[latex](\u2212,\u2212)[\/latex]<\/td>\r\n<td>[latex](+,\u2212)[\/latex]<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n<\/li>\r\n \t<li><strong>Coordinates of Zero<\/strong>\r\n<ul>\r\n \t<li>Points with a [latex]y[\/latex]-coordinate equal to [latex]0[\/latex] are on the <em>x-<\/em>axis, and have coordinates [latex] (a, 0)[\/latex].<\/li>\r\n \t<li>Points with a [latex]x[\/latex]-coordinate equal to [latex]0[\/latex] are on the <em>y-<\/em>axis, and have coordinates [latex](0, b)[\/latex].<\/li>\r\n \t<li>The point [latex](0, 0)[\/latex] is called the origin. It is the point where the <em>x-<\/em>axis and <em>y-<\/em>axis intersect.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Intercepts<\/strong>\r\n<ul>\r\n \t<li>The [latex]x[\/latex]-intercept is the point, [latex]\\left(a,0\\right)[\/latex] , where the graph crosses the [latex]x[\/latex]-axis. The [latex]x[\/latex]-intercept occurs when [latex]y[\/latex] is zero.<\/li>\r\n \t<li>The [latex]y[\/latex]-intercept is the point, [latex]\\left(0,b\\right)[\/latex] , where the graph crosses the [latex]y[\/latex]-axis. The [latex]y[\/latex]-intercept occurs when [latex]y[\/latex] is zero.<\/li>\r\n \t<li>The [latex]x[\/latex]-intercept occurs when [latex]y[\/latex] is zero.<\/li>\r\n \t<li>The [latex]y[\/latex]-intercept occurs when [latex]x[\/latex] is zero.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Find the <em>x<\/em> and <em>y<\/em> intercepts from the equation of a line<\/strong>\r\n<ul>\r\n \t<li>To find the [latex]x[\/latex]-intercept of the line, let [latex]y=0[\/latex] and solve for [latex]x[\/latex].<\/li>\r\n \t<li>To find the [latex]y[\/latex]-intercept of the line, let [latex]x=0[\/latex] and solve for [latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Graph a line using the intercepts<\/strong>\r\n<ol class=\"stepwise\">\r\n \t<li>Find the <em>x-<\/em> and <em>y-<\/em> intercepts of the line.\r\n<ul>\r\n \t<li>Let [latex]y=0[\/latex] and solve for [latex]x[\/latex].<\/li>\r\n \t<li>Let [latex]x=0[\/latex] and solve for [latex]y[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Find a third solution to the equation.<\/li>\r\n \t<li>Plot the three points and then check that they line up.<\/li>\r\n \t<li>Draw the line.<\/li>\r\n<\/ol>\r\n<\/li>\r\n \t<li><strong>Choose the most convenient method to graph a line<\/strong><\/li>\r\n<\/ul>\r\n<ol class=\"stepwise\">\r\n \t<li>Determine if the equation has only one variable. Then it is a vertical or horizontal line.\r\n<ul>\r\n \t<li>[latex]x=a[\/latex] is a vertical line passing through the [latex]x[\/latex]-axis at [latex]a[\/latex].<\/li>\r\n \t<li>[latex]y=b[\/latex] is a vertical line passing through the [latex]y[\/latex]-axis at [latex]b[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li>Determine if y is isolated on one side of the equation. The graph by plotting points.Choose any three values for <em>x<\/em> and then solve for the corresponding <em>y-<\/em> values.<\/li>\r\n \t<li>Determine if the equation is of the form [latex]Ax+By=C[\/latex] , find the intercepts.Find the <em>x-<\/em> and <em>y-<\/em> intercepts and then a third point.<\/li>\r\n<\/ol>\r\n<strong>Find the slope from a graph<\/strong>\r\n<ol id=\"eip-id1170322684949\" class=\"stepwise\">\r\n \t<li>Locate two points on the line whose coordinates are integers.<\/li>\r\n \t<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\r\n \t<li>Count the rise and the run on the legs of the triangle.<\/li>\r\n \t<li>Take the ratio of rise to run to find the slope, [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/li>\r\n<\/ol>\r\n<ul id=\"eip-678\">\r\n \t<li><strong>Slope of a Horizontal Line<\/strong>\r\n<ul id=\"eip-id1170320559493\">\r\n \t<li>The slope of a horizontal line, [latex]y=b[\/latex] , is [latex]0[\/latex].<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Slope of a Vertical Line<\/strong>\r\n<ul id=\"eip-id1170323909195\">\r\n \t<li>The slope of a vertical line, [latex]x=a[\/latex] , is undefined.<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Slope Formula<\/strong>\r\n<ul id=\"eip-id1170322772650\">\r\n \t<li>The slope of the line between two points [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] is [latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/li>\r\n<\/ul>\r\n<\/li>\r\n \t<li><strong>Graph a line given a point and a slope.<\/strong>\r\n<ol id=\"eip-id1170326457932\" class=\"stepwise\">\r\n \t<li>Plot the given point.<\/li>\r\n \t<li>Use the slope formula to identify the rise and the run.<\/li>\r\n \t<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\r\n \t<li>Connect the points with a line.<\/li>\r\n<\/ol>\r\n<\/li>\r\n<\/ul>\r\n<h2>Key Equations<\/h2>\r\n<ul>\r\n \t<li>The slope of the line between two points [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] is [latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex].<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl>\r\n \t<dt><strong>intercepts of a line<\/strong><\/dt>\r\n \t<dd>Each of the points at which a line crosses the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis is called an intercept of the line.<\/dd>\r\n \t<dt>linear equation<\/dt>\r\n \t<dd>An equation of the form [latex]Ax+By=C[\/latex], where [latex]A[\/latex] and [latex]B[\/latex] are not both zero, is called a linear equation in two variables.<\/dd>\r\n \t<dt>ordered pair<\/dt>\r\n \t<dd>An ordered pair [latex]\\left(x,y\\right)[\/latex] gives the coordinates of a point in a rectangular coordinate system. The first number is the [latex]x[\/latex] -coordinate. The second number is the [latex]y[\/latex] -coordinate.<\/dd>\r\n \t<dt>origin<\/dt>\r\n \t<dd>The point [latex]\\left(0,0\\right)[\/latex] is called the origin. It is the point where the the point where the [latex]x[\/latex] -axis and [latex]y[\/latex] -axis intersect.<\/dd>\r\n \t<dt>quadrants<\/dt>\r\n \t<dd>The [latex]x[\/latex] -axis and [latex]y[\/latex] -axis divide a rectangular coordinate system into four areas, called quadrants.<\/dd>\r\n \t<dt>slope of a line<\/dt>\r\n \t<dd>The slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex] . The rise measures the vertical change and the run measures the horizontal change.<\/dd>\r\n \t<dt>solution to a linear equation in two variables<\/dt>\r\n \t<dd>An ordered pair [latex]\\left(x,y\\right)[\/latex] is a solution to the linear equation [latex]Ax+By=C[\/latex], if the equation is a true statement when the <em>x-<\/em> and <em>y<\/em>-values of the ordered pair are substituted into the equation.<\/dd>\r\n \t<dt><em>x<\/em>-axis<\/dt>\r\n \t<dd>The <em>x<\/em>-axis is the horizontal axis in a rectangular coordinate system.<\/dd>\r\n \t<dt><em>y<\/em>-axis<\/dt>\r\n \t<dd>The <em>y<\/em>-axis is the vertical axis on a rectangular coordinate system.<\/dd>\r\n<\/dl>\r\n<dl>\r\n \t<dt><strong>term<\/strong><\/dt>\r\n \t<dd>definition<\/dd>\r\n<\/dl>\r\n<dl>\r\n \t<dt><strong>term<\/strong><\/dt>\r\n \t<dd>definition<\/dd>\r\n<\/dl>","rendered":"<h2>Key Concepts<\/h2>\n<ul>\n<li><strong>Sign Patterns of the Quadrants<\/strong><br \/>\n<table id=\"eip-id1170324021306\" class=\"unnumbered unstyled\" summary=\"...\">\n<thead>\n<tr>\n<th>Quadrant I<\/th>\n<th>Quadrant II<\/th>\n<th>Quadrant III<\/th>\n<th>Quadrant IV<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<td>[latex](x,y)[\/latex]<\/td>\n<td>[latex](x,y)[\/latex]<\/td>\n<td>[latex](x,y)[\/latex]<\/td>\n<td>[latex](x,y)[\/latex]<\/td>\n<\/tr>\n<tr>\n<td>[latex](+,+)[\/latex]<\/td>\n<td>[latex](\u2212,+)[\/latex]<\/td>\n<td>[latex](\u2212,\u2212)[\/latex]<\/td>\n<td>[latex](+,\u2212)[\/latex]<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/li>\n<li><strong>Coordinates of Zero<\/strong>\n<ul>\n<li>Points with a [latex]y[\/latex]-coordinate equal to [latex]0[\/latex] are on the <em>x-<\/em>axis, and have coordinates [latex](a, 0)[\/latex].<\/li>\n<li>Points with a [latex]x[\/latex]-coordinate equal to [latex]0[\/latex] are on the <em>y-<\/em>axis, and have coordinates [latex](0, b)[\/latex].<\/li>\n<li>The point [latex](0, 0)[\/latex] is called the origin. It is the point where the <em>x-<\/em>axis and <em>y-<\/em>axis intersect.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Intercepts<\/strong>\n<ul>\n<li>The [latex]x[\/latex]-intercept is the point, [latex]\\left(a,0\\right)[\/latex] , where the graph crosses the [latex]x[\/latex]-axis. The [latex]x[\/latex]-intercept occurs when [latex]y[\/latex] is zero.<\/li>\n<li>The [latex]y[\/latex]-intercept is the point, [latex]\\left(0,b\\right)[\/latex] , where the graph crosses the [latex]y[\/latex]-axis. The [latex]y[\/latex]-intercept occurs when [latex]y[\/latex] is zero.<\/li>\n<li>The [latex]x[\/latex]-intercept occurs when [latex]y[\/latex] is zero.<\/li>\n<li>The [latex]y[\/latex]-intercept occurs when [latex]x[\/latex] is zero.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Find the <em>x<\/em> and <em>y<\/em> intercepts from the equation of a line<\/strong>\n<ul>\n<li>To find the [latex]x[\/latex]-intercept of the line, let [latex]y=0[\/latex] and solve for [latex]x[\/latex].<\/li>\n<li>To find the [latex]y[\/latex]-intercept of the line, let [latex]x=0[\/latex] and solve for [latex]y[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Graph a line using the intercepts<\/strong>\n<ol class=\"stepwise\">\n<li>Find the <em>x-<\/em> and <em>y-<\/em> intercepts of the line.\n<ul>\n<li>Let [latex]y=0[\/latex] and solve for [latex]x[\/latex].<\/li>\n<li>Let [latex]x=0[\/latex] and solve for [latex]y[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Find a third solution to the equation.<\/li>\n<li>Plot the three points and then check that they line up.<\/li>\n<li>Draw the line.<\/li>\n<\/ol>\n<\/li>\n<li><strong>Choose the most convenient method to graph a line<\/strong><\/li>\n<\/ul>\n<ol class=\"stepwise\">\n<li>Determine if the equation has only one variable. Then it is a vertical or horizontal line.\n<ul>\n<li>[latex]x=a[\/latex] is a vertical line passing through the [latex]x[\/latex]-axis at [latex]a[\/latex].<\/li>\n<li>[latex]y=b[\/latex] is a vertical line passing through the [latex]y[\/latex]-axis at [latex]b[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li>Determine if y is isolated on one side of the equation. The graph by plotting points.Choose any three values for <em>x<\/em> and then solve for the corresponding <em>y-<\/em> values.<\/li>\n<li>Determine if the equation is of the form [latex]Ax+By=C[\/latex] , find the intercepts.Find the <em>x-<\/em> and <em>y-<\/em> intercepts and then a third point.<\/li>\n<\/ol>\n<p><strong>Find the slope from a graph<\/strong><\/p>\n<ol id=\"eip-id1170322684949\" class=\"stepwise\">\n<li>Locate two points on the line whose coordinates are integers.<\/li>\n<li>Starting with the point on the left, sketch a right triangle, going from the first point to the second point.<\/li>\n<li>Count the rise and the run on the legs of the triangle.<\/li>\n<li>Take the ratio of rise to run to find the slope, [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex]<\/li>\n<\/ol>\n<ul id=\"eip-678\">\n<li><strong>Slope of a Horizontal Line<\/strong>\n<ul id=\"eip-id1170320559493\">\n<li>The slope of a horizontal line, [latex]y=b[\/latex] , is [latex]0[\/latex].<\/li>\n<\/ul>\n<\/li>\n<li><strong>Slope of a Vertical Line<\/strong>\n<ul id=\"eip-id1170323909195\">\n<li>The slope of a vertical line, [latex]x=a[\/latex] , is undefined.<\/li>\n<\/ul>\n<\/li>\n<li><strong>Slope Formula<\/strong>\n<ul id=\"eip-id1170322772650\">\n<li>The slope of the line between two points [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] is [latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex]<\/li>\n<\/ul>\n<\/li>\n<li><strong>Graph a line given a point and a slope.<\/strong>\n<ol id=\"eip-id1170326457932\" class=\"stepwise\">\n<li>Plot the given point.<\/li>\n<li>Use the slope formula to identify the rise and the run.<\/li>\n<li>Starting at the given point, count out the rise and run to mark the second point.<\/li>\n<li>Connect the points with a line.<\/li>\n<\/ol>\n<\/li>\n<\/ul>\n<h2>Key Equations<\/h2>\n<ul>\n<li>The slope of the line between two points [latex]\\left({x}_{1},{y}_{1}\\right)[\/latex] and [latex]\\left({x}_{2},{y}_{2}\\right)[\/latex] is [latex]m={\\Large\\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}}[\/latex].<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl>\n<dt><strong>intercepts of a line<\/strong><\/dt>\n<dd>Each of the points at which a line crosses the [latex]x[\/latex]-axis and the [latex]y[\/latex]-axis is called an intercept of the line.<\/dd>\n<dt>linear equation<\/dt>\n<dd>An equation of the form [latex]Ax+By=C[\/latex], where [latex]A[\/latex] and [latex]B[\/latex] are not both zero, is called a linear equation in two variables.<\/dd>\n<dt>ordered pair<\/dt>\n<dd>An ordered pair [latex]\\left(x,y\\right)[\/latex] gives the coordinates of a point in a rectangular coordinate system. The first number is the [latex]x[\/latex] -coordinate. The second number is the [latex]y[\/latex] -coordinate.<\/dd>\n<dt>origin<\/dt>\n<dd>The point [latex]\\left(0,0\\right)[\/latex] is called the origin. It is the point where the the point where the [latex]x[\/latex] -axis and [latex]y[\/latex] -axis intersect.<\/dd>\n<dt>quadrants<\/dt>\n<dd>The [latex]x[\/latex] -axis and [latex]y[\/latex] -axis divide a rectangular coordinate system into four areas, called quadrants.<\/dd>\n<dt>slope of a line<\/dt>\n<dd>The slope of a line is [latex]m={\\Large\\frac{\\text{rise}}{\\text{run}}}[\/latex] . The rise measures the vertical change and the run measures the horizontal change.<\/dd>\n<dt>solution to a linear equation in two variables<\/dt>\n<dd>An ordered pair [latex]\\left(x,y\\right)[\/latex] is a solution to the linear equation [latex]Ax+By=C[\/latex], if the equation is a true statement when the <em>x-<\/em> and <em>y<\/em>-values of the ordered pair are substituted into the equation.<\/dd>\n<dt><em>x<\/em>-axis<\/dt>\n<dd>The <em>x<\/em>-axis is the horizontal axis in a rectangular coordinate system.<\/dd>\n<dt><em>y<\/em>-axis<\/dt>\n<dd>The <em>y<\/em>-axis is the vertical axis on a rectangular coordinate system.<\/dd>\n<\/dl>\n<dl>\n<dt><strong>term<\/strong><\/dt>\n<dd>definition<\/dd>\n<\/dl>\n<dl>\n<dt><strong>term<\/strong><\/dt>\n<dd>definition<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-3089\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Original<\/div><ul class=\"citation-list\"><li><strong>Authored by<\/strong>: Deborah Devlin. <strong>Provided by<\/strong>: Lumen Learning. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by\/4.0\/\">CC BY: Attribution<\/a><\/em><\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t <\/section>","protected":false},"author":25777,"menu_order":13,"template":"","meta":{"_candela_citation":"[{\"type\":\"original\",\"description\":\"\",\"author\":\"Deborah Devlin\",\"organization\":\"Lumen Learning\",\"url\":\"\",\"project\":\"\",\"license\":\"cc-by\",\"license_terms\":\"\"}]","CANDELA_OUTCOMES_GUID":"","pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":""},"chapter-type":[],"contributor":[],"license":[],"class_list":["post-3089","chapter","type-chapter","status-web-only","hentry"],"part":1040,"_links":{"self":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3089","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/users\/25777"}],"version-history":[{"count":4,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3089\/revisions"}],"predecessor-version":[{"id":3802,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3089\/revisions\/3802"}],"part":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/parts\/1040"}],"metadata":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapters\/3089\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/media?parent=3089"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/pressbooks\/v2\/chapter-type?post=3089"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/contributor?post=3089"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/courses.lumenlearning.com\/slcc-mathforliberalartscorequisite\/wp-json\/wp\/v2\/license?post=3089"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}