{"id":17664,"date":"2021-08-19T16:09:37","date_gmt":"2021-08-19T16:09:37","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/?post_type=chapter&#038;p=17664"},"modified":"2021-08-19T16:37:24","modified_gmt":"2021-08-19T16:37:24","slug":"section-1-2-graphs-of-equations-in-two-variables-intercepts-symmetry","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/csn-precalculusv2\/chapter\/section-1-2-graphs-of-equations-in-two-variables-intercepts-symmetry\/","title":{"raw":"Section 1.2: Graphs of Equations in Two Variables; Intercepts; Symmetry","rendered":"Section 1.2: Graphs of Equations in Two Variables; Intercepts; Symmetry"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Learning Outcomes<\/h3>\r\n<ul>\r\n \t<li>Find x- and y-intercepts.<\/li>\r\n \t<li>Use intercepts to plot lines.<\/li>\r\n \t<li>Test for x-axis, y-axis, and origin symmetry.<\/li>\r\n<\/ul>\r\n<\/div>\r\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\r\nThe <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.\r\n\r\nTo determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].\r\n\r\nTo find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill &amp; \\hfill \\\\ 0=3x - 1\\hfill &amp; \\hfill \\\\ 1=3x\\hfill &amp; \\hfill \\\\ \\frac{1}{3}=x\\hfill &amp; \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill &amp; x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nTo find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill &amp; \\hfill \\\\ y=3\\left(0\\right)-1\\hfill &amp; \\hfill \\\\ y=-1\\hfill &amp; \\hfill \\\\ \\left(0,-1\\right)\\hfill &amp; y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nWe can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/>\r\n<div class=\"textbox\">\r\n<h3>How To: Given an equation, find the intercepts<\/h3>\r\n<ol>\r\n \t<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\r\n \t<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div class=\"textbox exercises\">\r\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\r\nFind the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.\r\n\r\n[reveal-answer q=\"814560\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"814560\"]\r\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nSet [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.\r\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\r\nPlot both points and draw a line passing through them.\r\n\r\n<img class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\nFind the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].\r\n\r\n[reveal-answer q=\"80464\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"80464\"]\r\n\r\n<em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].\r\n\r\n<img class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/>\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<div class=\"textbox key-takeaways\">\r\n<h3>Try It<\/h3>\r\n[ohm_question]92757[\/ohm_question]\r\n\r\n<\/div>\r\n<h2>Symmetry<\/h2>\r\nThink of symmetry as a fold line. If a graph can be folded on top of itself and everything overlaps, then it has <strong>symmetry<\/strong>. The fold line that allows this to happen is called the <strong>line of symmetry<\/strong>.\r\n<div class=\"textbox\">\r\n<h3>Symmetry<\/h3>\r\nBelow are the three types of symmetry:\r\n<img src=\"http:\/\/www.hutchmath.com\/Images\/symmetry.JPG\" \/>\r\n\r\n<\/div>\r\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\r\n<h3 id=\"fs-id1165137762240\">How To: test for symmetry without graphing<\/h3>\r\n<ol id=\"fs-id1165137442714\">\r\n \t<li>x-axis symmetry: Replace y by -y in the original equation. If it simplifies to the original equation it has this symmetry.<\/li>\r\n \t<li>y-axis symmetry: Replace x by -x in the original equation. If it simplifies to the original equation it has this symmetry.<\/li>\r\n \t<li>origin symmetry: Replace y by -y and x by -x in the original equation. If it simplifies to the original equation it has this symmetry.<\/li>\r\n<\/ol>\r\n<\/div>\r\n<div id=\"Example_01_03_06\" class=\"example\">\r\n<div id=\"fs-id1165135174952\" class=\"exercise\">\r\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\r\n<h3>Example 9: Testing for Symmetry Algebraically<\/h3>\r\n<p id=\"fs-id1165135155397\">Test the following equation for symmetry: [latex]y^2=3x-4[\/latex]<\/p>\r\n[reveal-answer q=\"138600\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"138600\"]\r\nWe will go through each test separately:\r\n\r\n<strong>x-axis symmetry:<\/strong> [latex]\\left(-y\\right)^2=3x-4[\/latex] which simplifies to [latex]y^2=3x-4[\/latex].\r\n\r\nThis is the same equation as the original so this equation has x-axis symmetry.\r\n\r\n<strong>y-axis symmetry:<\/strong> [latex]y^2=3(-x)-4[\/latex] which simplifies to [latex]y^2=-3x-4[\/latex].\r\n\r\nThis is the not the same equation as the original so this equation does not have x-axis symmetry.\r\n\r\n<strong>origin symmetry:<\/strong> [latex]\\left(-y\\right)^2=3(-x)-4[\/latex] which simplifies to [latex]y^2=-3x-4[\/latex].\r\n\r\nThis is the not the same equation as the original so this equation does not have origin symmetry.\r\n\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"bcc-box bcc-success\">\r\n<h3>Try It<\/h3>\r\n<p id=\"fs-id1165134149846\">Test the following equation for symmetry: [latex]y=2x^3-x[\/latex].<\/p>\r\n[reveal-answer q=\"944500\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"944500\"]\r\nOrigin symmetry\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<dl id=\"fs-id1165135315542\" class=\"definition\">\r\n \t<dd><\/dd>\r\n<\/dl>\r\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 1.2 Homework Exercises<\/span><\/h2>\r\nFor each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.\r\n\r\n1.\u00a0 [latex]y=-3x+6[\/latex]\r\n\r\n2.\u00a0 [latex]4y=2x-1[\/latex]\r\n\r\n3.\u00a0 [latex]3x-2y=6[\/latex]\r\n\r\n4.\u00a0 [latex]4x-3=2y[\/latex]\r\n\r\n5.\u00a0 [latex]3x+8y=9[\/latex]\r\n\r\n6. [latex]2x-\\frac{2}{3}=\\frac{3}{4}y+3[\/latex]\r\n\r\nFor each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.\r\n\r\n7.\u00a0 [latex]x-2y=8[\/latex]\r\n\r\n8.\u00a0 [latex]y-5=5x[\/latex]\r\n\r\n9.\u00a0 [latex]3y=-2x+6[\/latex]\r\n\r\n10.\u00a0 [latex]y=\\frac{x-3}{2}[\/latex]\r\n\r\nFor the following exercises, test for symmetry.\r\n\r\n11. [latex]y=x^2+12[\/latex]\r\n\r\n12. [latex]x=y^2+1[\/latex]\r\n\r\n13. [latex]x=y^2-5[\/latex]\r\n\r\n14. [latex]y^2=x^2+4[\/latex]\r\n\r\n15. [latex]y^2=x^2-7[\/latex]\r\n\r\n16. [latex]x^2+y^3=3[\/latex]\r\n\r\n17. [latex]x^3-y^2=11[\/latex]\r\n\r\n18. [latex]y=2x-3[\/latex]","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Learning Outcomes<\/h3>\n<ul>\n<li>Find x- and y-intercepts.<\/li>\n<li>Use intercepts to plot lines.<\/li>\n<li>Test for x-axis, y-axis, and origin symmetry.<\/li>\n<\/ul>\n<\/div>\n<h2>Using Intercepts to Plot Lines in the Coordinate Plane<\/h2>\n<p>The <strong>intercepts<\/strong> of a graph are points where the graph crosses the axes. The <strong><em>x-<\/em>intercept<\/strong> is the point where the graph crosses the <em>x-<\/em>axis. At this point, the <em>y-<\/em>coordinate is zero. The <strong><em>y-<\/em>intercept<\/strong> is the point where the graph crosses the <em>y-<\/em>axis. At this point, the <em>x-<\/em>coordinate is zero.<\/p>\n<p>To determine the <em>x-<\/em>intercept, we set <em>y <\/em>equal to zero and solve for <em>x<\/em>. Similarly, to determine the <em>y-<\/em>intercept, we set <em>x <\/em>equal to zero and solve for <em>y<\/em>. For example, lets find the intercepts of the equation [latex]y=3x - 1[\/latex].<\/p>\n<p>To find the <em>x-<\/em>intercept, set [latex]y=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{llllll}y=3x - 1\\hfill & \\hfill \\\\ 0=3x - 1\\hfill & \\hfill \\\\ 1=3x\\hfill & \\hfill \\\\ \\frac{1}{3}=x\\hfill & \\hfill \\\\ \\left(\\frac{1}{3},0\\right)\\hfill & x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>To find the <em>y-<\/em>intercept, set [latex]x=0[\/latex].<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{lllll}y=3x - 1\\hfill & \\hfill \\\\ y=3\\left(0\\right)-1\\hfill & \\hfill \\\\ y=-1\\hfill & \\hfill \\\\ \\left(0,-1\\right)\\hfill & y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>We can confirm that our results make sense by observing a graph of the equation. Notice that the graph crosses the axes where we predicted it would.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042423\/CNX_CAT_Figure_02_01_012.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y-axis range from negative 4 to 4. The function y = 3x \u2013 1 is plotted on the coordinate plane\" width=\"487\" height=\"366\" \/><\/p>\n<div class=\"textbox\">\n<h3>How To: Given an equation, find the intercepts<\/h3>\n<ol>\n<li>Find the <em>x<\/em>-intercept by setting [latex]y=0[\/latex] and solving for [latex]x[\/latex].<\/li>\n<li>Find the <em>y-<\/em>intercept by setting [latex]x=0[\/latex] and solving for [latex]y[\/latex].<\/li>\n<\/ol>\n<\/div>\n<div class=\"textbox exercises\">\n<h3>Example: Finding the Intercepts of the Given Equation<\/h3>\n<p>Find the intercepts of the equation [latex]y=-3x - 4[\/latex]. Then sketch the graph using only the intercepts.<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q814560\">Show Solution<\/span><\/p>\n<div id=\"q814560\" class=\"hidden-answer\" style=\"display: none\">\nSet [latex]y=0[\/latex] to find the <em>x-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ 0=-3x - 4\\hfill \\\\ 4=-3x\\hfill \\\\ -\\frac{4}{3}=x\\hfill \\\\ \\left(-\\frac{4}{3},0\\right)x\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Set [latex]x=0[\/latex] to find the <em>y-<\/em>intercept.<\/p>\n<div style=\"text-align: center;\">[latex]\\begin{array}{l}y=-3x - 4\\hfill \\\\ y=-3\\left(0\\right)-4\\hfill \\\\ y=-4\\hfill \\\\ \\left(0,-4\\right)y\\text{-intercept}\\hfill \\end{array}[\/latex]<\/div>\n<p>Plot both points and draw a line passing through them.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images\/wp-content\/uploads\/sites\/896\/2016\/10\/12042425\/CNX_CAT_Figure_02_01_013.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x-axis ranges from negative 5 to 5. The y-axis ranges from negative 6 to 3. The line passes through the points (-4\/3, 0) and (0, -4).\" width=\"487\" height=\"406\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p>Find the intercepts of the equation and sketch the graph: [latex]y=-\\frac{3}{4}x+3[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q80464\">Show Solution<\/span><\/p>\n<div id=\"q80464\" class=\"hidden-answer\" style=\"display: none\">\n<p><em>x<\/em>-intercept is [latex]\\left(4,0\\right)[\/latex]; <em>y-<\/em>intercept is [latex]\\left(0,3\\right)[\/latex].<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"alignnone\" src=\"https:\/\/s3-us-west-2.amazonaws.com\/courses-images-archive-read-only\/wp-content\/uploads\/sites\/924\/2015\/09\/25200257\/CNX_CAT_Figure_02_01_014.jpg\" alt=\"This is an image of a line graph on an x, y coordinate plane. The x and y axes range from negative 4 to 6. The function y = -3x\/4 + 3 is plotted.\" width=\"487\" height=\"447\" \/><\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"textbox key-takeaways\">\n<h3>Try It<\/h3>\n<p><iframe loading=\"lazy\" id=\"ohm92757\" class=\"resizable\" src=\"https:\/\/ohm.lumenlearning.com\/multiembedq.php?id=92757&theme=oea&iframe_resize_id=ohm92757&show_question_numbers\" width=\"100%\" height=\"150\"><\/iframe><\/p>\n<\/div>\n<h2>Symmetry<\/h2>\n<p>Think of symmetry as a fold line. If a graph can be folded on top of itself and everything overlaps, then it has <strong>symmetry<\/strong>. The fold line that allows this to happen is called the <strong>line of symmetry<\/strong>.<\/p>\n<div class=\"textbox\">\n<h3>Symmetry<\/h3>\n<p>Below are the three types of symmetry:<br \/>\n<img decoding=\"async\" src=\"http:\/\/www.hutchmath.com\/Images\/symmetry.JPG\" alt=\"image\" \/><\/p>\n<\/div>\n<div id=\"fs-id1165135530407\" class=\"note precalculus howto textbox\">\n<h3 id=\"fs-id1165137762240\">How To: test for symmetry without graphing<\/h3>\n<ol id=\"fs-id1165137442714\">\n<li>x-axis symmetry: Replace y by -y in the original equation. If it simplifies to the original equation it has this symmetry.<\/li>\n<li>y-axis symmetry: Replace x by -x in the original equation. If it simplifies to the original equation it has this symmetry.<\/li>\n<li>origin symmetry: Replace y by -y and x by -x in the original equation. If it simplifies to the original equation it has this symmetry.<\/li>\n<\/ol>\n<\/div>\n<div id=\"Example_01_03_06\" class=\"example\">\n<div id=\"fs-id1165135174952\" class=\"exercise\">\n<div id=\"fs-id1165135174954\" class=\"problem textbox shaded\">\n<h3>Example 9: Testing for Symmetry Algebraically<\/h3>\n<p id=\"fs-id1165135155397\">Test the following equation for symmetry: [latex]y^2=3x-4[\/latex]<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q138600\">Show Solution<\/span><\/p>\n<div id=\"q138600\" class=\"hidden-answer\" style=\"display: none\">\nWe will go through each test separately:<\/p>\n<p><strong>x-axis symmetry:<\/strong> [latex]\\left(-y\\right)^2=3x-4[\/latex] which simplifies to [latex]y^2=3x-4[\/latex].<\/p>\n<p>This is the same equation as the original so this equation has x-axis symmetry.<\/p>\n<p><strong>y-axis symmetry:<\/strong> [latex]y^2=3(-x)-4[\/latex] which simplifies to [latex]y^2=-3x-4[\/latex].<\/p>\n<p>This is the not the same equation as the original so this equation does not have x-axis symmetry.<\/p>\n<p><strong>origin symmetry:<\/strong> [latex]\\left(-y\\right)^2=3(-x)-4[\/latex] which simplifies to [latex]y^2=-3x-4[\/latex].<\/p>\n<p>This is the not the same equation as the original so this equation does not have origin symmetry.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"bcc-box bcc-success\">\n<h3>Try It<\/h3>\n<p id=\"fs-id1165134149846\">Test the following equation for symmetry: [latex]y=2x^3-x[\/latex].<\/p>\n<div class=\"qa-wrapper\" style=\"display: block\"><span class=\"show-answer collapsed\" style=\"cursor: pointer\" data-target=\"q944500\">Show Solution<\/span><\/p>\n<div id=\"q944500\" class=\"hidden-answer\" style=\"display: none\">\nOrigin symmetry\n<\/div>\n<\/div>\n<\/div>\n<dl id=\"fs-id1165135315542\" class=\"definition\">\n<dd><\/dd>\n<\/dl>\n<h2 style=\"text-align: center;\"><span style=\"text-decoration: underline;\">Section 1.2 Homework Exercises<\/span><\/h2>\n<p>For each of the following exercises, find the x-intercept and the y-intercept without graphing. Write the coordinates of each intercept.<\/p>\n<p>1.\u00a0 [latex]y=-3x+6[\/latex]<\/p>\n<p>2.\u00a0 [latex]4y=2x-1[\/latex]<\/p>\n<p>3.\u00a0 [latex]3x-2y=6[\/latex]<\/p>\n<p>4.\u00a0 [latex]4x-3=2y[\/latex]<\/p>\n<p>5.\u00a0 [latex]3x+8y=9[\/latex]<\/p>\n<p>6. [latex]2x-\\frac{2}{3}=\\frac{3}{4}y+3[\/latex]<\/p>\n<p>For each of the following exercises, find and plot the x- and y-intercepts, and graph the straight line based on those two points.<\/p>\n<p>7.\u00a0 [latex]x-2y=8[\/latex]<\/p>\n<p>8.\u00a0 [latex]y-5=5x[\/latex]<\/p>\n<p>9.\u00a0 [latex]3y=-2x+6[\/latex]<\/p>\n<p>10.\u00a0 [latex]y=\\frac{x-3}{2}[\/latex]<\/p>\n<p>For the following exercises, test for symmetry.<\/p>\n<p>11. [latex]y=x^2+12[\/latex]<\/p>\n<p>12. [latex]x=y^2+1[\/latex]<\/p>\n<p>13. [latex]x=y^2-5[\/latex]<\/p>\n<p>14. [latex]y^2=x^2+4[\/latex]<\/p>\n<p>15. [latex]y^2=x^2-7[\/latex]<\/p>\n<p>16. [latex]x^2+y^3=3[\/latex]<\/p>\n<p>17. [latex]x^3-y^2=11[\/latex]<\/p>\n<p>18. 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