{"id":4774,"date":"2021-08-25T15:34:42","date_gmt":"2021-08-25T15:34:42","guid":{"rendered":"https:\/\/mathemerize.com\/?p=4774"},"modified":"2022-01-16T16:57:53","modified_gmt":"2022-01-16T11:27:53","slug":"point-of-intersection-of-two-lines","status":"publish","type":"post","link":"https:\/\/mathemerize.com\/point-of-intersection-of-two-lines\/","title":{"rendered":"Point of Intersection of Two Lines – Formula and Example"},"content":{"rendered":"

Here you will learn how to find point of intersection of two lines with examples.<\/p>\n

Let’s begin –<\/p>\n

How to find Point of Intersection of Two Lines<\/h2>\n

Let the equations of two lines be<\/p>\n

\\(a_1x + b_1y + c_1\\) = 0<\/p>\n

and,  \\(a_2x + b_2y + c_2\\) = 0<\/p>\n

Suppose these two lines intersect at a point P(\\(x_1, y_1\\)). Then, (\\(x_1, y_1\\)) satisfies each of the given equations.<\/p>\n

\\(\\therefore\\)  \\(a_1x_1 + b_1y_1 + c_1\\) = 0   and   \\(a_2x_1 + b_2y_1 + c_2\\) = 0<\/p>\n

Solving these two by cross-multiplication, we get<\/p>\n

\\(x_1\\over {b_1c_2 – b_2c_1}\\) = \\(y_1\\over {c_1a_2 – c_2a_1}\\) = \\(1\\over {a_1b_2 – a_2b_1}\\)<\/p>\n

\\(\\implies\\)  \\(x_1\\) = \\({b_1c_2 – b_2c_1}\\over {a_1b_2 – a_2b_1}\\),  \\(y_1\\) = \\({c_1a_2 – c_2a_1}\\over {a_1b_2 – a_2b_1}\\)<\/p>\n

Hence the coordinates of the point of the point of intersection of two lines are :<\/p>\n

( \\({b_1c_2 – b_2c_1}\\over {a_1b_2 – a_2b_1}\\), \\({c_1a_2 – c_2a_1}\\over {a_1b_2 – a_2b_1}\\))<\/p>\n

Formula to find Point of Intersection :<\/strong><\/h4>\n
\n

\\(x_1\\) = ( \\({b_1c_2 – b_2c_1}\\over {a_1b_2 – a_2b_1}\\), \\(y_1\\) = \\({c_1a_2 – c_2a_1}\\over {a_1b_2 – a_2b_1}\\))<\/p>\n<\/blockquote>\n

Note : <\/strong>To find the coordinates of the point of intersection of two non-parallel lines, we solve the given equations simultaneously and the values of x and y are so obtained determine the coordinates of the point of intersection.<\/p>\n\n\n

Example : <\/span>Find the coordinates of the point of intersecton of the lines 2x – y + 3 = 0 and x + 2y – 4 = 0.<\/p>\n

Solution : <\/span>Solving simultaneously the equations 2x – y + 3 = 0 and x + 2y – 4 = 0, we obtain

\n\\(x\\over {4-6}\\) = \\(y\\over {3+8}\\) = \\(1\\over {4+1}\\)

\n\\(\\implies\\) \\(x\\over -2\\) = \\(y\\over 11\\) = \\(1\\over 5\\)

\n\\(\\implies\\) x = \\(-2\\over 5\\) , y = \\(11\\over 5\\)

\nHence, (-2\/5, 11\/5) is the required point of intersection

<\/p>\n\n\n\n

Example : <\/span>Find the coordinates of the point of intersecton of the lines x – y + 4 = 0 and x + 2y – 1 = 0.<\/p>\n

Solution : <\/span>Solving simultaneously the equations x – y + 4 = 0 and x + 2y – 1 = 0, we obtain

\n\\(x\\over {1-8}\\) = \\(y\\over {4+1}\\) = \\(1\\over {2+1}\\)

\n\\(\\implies\\) \\(x\\over -7\\) = \\(y\\over 5\\) = \\(1\\over 3\\)

\n\\(\\implies\\) x = \\(-7\\over 3\\) , y = \\(5\\over 3\\)

\nHence, (-7\/3, 5\/3) is the required point of intersection
<\/p>\n\n\n

\n

Related Questions<\/h3>\n

Find the equation of line joining the point (3, 5) to the point of intersection of the lines 4x + y \u2013 1 = 0 and 7x \u2013 3y \u2013 35 = 0.<\/a><\/p>\n

Find the coordinates of the point of intersecton of the lines 2x \u2013 y + 3 = 0 and x + y \u2013 5 = 0.<\/a><\/p>\n

Find the equation of line parallel to y-axis and drawn through the point of intersection of the lines x \u2013 7y + 5 = 0 and 3x + y = 0.<\/a><\/p>\n\n\n

\n
Previous – Family of Lines \u2013 The combined Equation of Angle Bisectors<\/a><\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Here you will learn how to find point of intersection of two lines with examples. Let’s begin – How to find Point of Intersection of Two Lines Let the equations of two lines be \\(a_1x + b_1y + c_1\\) = 0 and,  \\(a_2x + b_2y + c_2\\) = 0 Suppose these two lines intersect at …<\/p>\n

Point of Intersection of Two Lines – Formula and Example<\/span> Read More »<\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"default","ast-global-header-display":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","theme-transparent-header-meta":"default","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":""},"categories":[32],"tags":[637,636,635],"yoast_head":"\nPoint of Intersection of Two Lines - Formula and Example<\/title>\n<meta name=\"description\" content=\"In this post you will learn how to find point of intersection of two lines with examples.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/mathemerize.com\/point-of-intersection-of-two-lines\/\" \/>\n<meta property=\"og:locale\" content=\"en_US\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Point of Intersection of Two Lines - 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