The point of intersection of two or more lines is a point which lies on all the given lies. It means the equations of all the given lines must be satisfied by the intersection point. This point of intersection of lines is called the \u201cpoint of concurrency\u201d.\u00a0 Finding this point of concurrency of two lines from given set of lines is used to determine whether the other lines are concurrent with these two lines.<\/p>\n
Let two lines a1<\/sub>x+b1<\/sub>y+c1<\/sub> =0<\/strong> and a2<\/sub>x + b2<\/sub>y + c2<\/sub>=0<\/strong> represent two intersecting lines.<\/p>\n Let the intersecting point of these two lines be (x1<\/sub>,y1<\/sub>).<\/strong><\/p>\n Then it must lie on both the lines.<\/p>\n a1<\/sub>x1<\/sub>+b1<\/sub>y1<\/sub>+c1<\/sub>=0<\/strong><\/p>\n a2<\/sub>x1<\/sub>+b2<\/sub>y1<\/sub>+c2<\/sub>=0<\/strong><\/p>\n Solving these two equation, we get<\/p>\n (x1<\/sub>,y1<\/sub>)=([b1<\/sub>c2<\/sub>-b2<\/sub>c1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>], [c1<\/sub>a2<\/sub>-c2<\/sub>a1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>])<\/strong><\/p>\n Example:<\/strong><\/p>\n Find the point of intersection of lines 2x+3y-8=0<\/strong> and x-y +1=0.<\/strong><\/p>\n Solution:<\/strong><\/p>\n Here a1<\/sub>=2, b1<\/sub>=3, c1<\/sub>=-8 and a2<\/sub>=1, b2<\/sub>=-1, c2<\/sub>=1,<\/strong><\/p>\n Pint of intersection (x,y) = ([b1<\/sub>c2<\/sub>-b2<\/sub>c1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>], [c1<\/sub>a2<\/sub>-c2<\/sub>a1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>])<\/strong><\/p>\n =([3-8]\/[-2-3], [-8-2]\/ [-2-3])=(-5\/-5, -10\/-5)=(1,2).<\/strong><\/p>\n Let the 3<\/strong> lines a1<\/sub>x+b1<\/sub>y+c1<\/sub> =0, a2<\/sub>x + b2<\/sub>y + c2<\/sub>=0<\/strong> and a3<\/sub>x + b3<\/sub>y + c3<\/sub>=0<\/strong> be concurrent. Then the point of intersection of first two lines must lie on the third line.<\/p>\n i.e. ([b1<\/sub>c2<\/sub>-b2<\/sub>c1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>], [c1<\/sub>a2<\/sub>-c2<\/sub>a1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>])<\/strong> must line on the line a3<\/sub>x + b3<\/sub>y + c3<\/sub>=0.<\/strong><\/p>\n So, a3<\/sub> [b1<\/sub>c2<\/sub>-b2<\/sub>c1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>] \u00a0+ b3 <\/sub>[c1<\/sub>a2<\/sub>-c2<\/sub>a1<\/sub>]\/[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>] \u00a0+ c3 <\/sub>= 0<\/strong><\/p>\n Then we get<\/p>\n a3<\/sub> [b1<\/sub>c2<\/sub>-b2<\/sub>c1<\/sub>] + b3 <\/sub>[c1<\/sub>a2<\/sub>-c2<\/sub>a1<\/sub>] + c3 <\/sub>[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>] = 0.<\/strong><\/p>\n Example:<\/strong><\/p>\n Determine whether the lines 2x+3y-8=0 , x -y +1=0<\/strong> and 3x + y-5=0.<\/strong><\/p>\n Solution:<\/strong><\/em><\/p>\n a1<\/sub>=2, b1<\/sub>=3, c1<\/sub>=-8 , a2<\/sub>=1, b2<\/sub>=-1, c2<\/sub>=1<\/strong> and a3<\/sub>=3, b3<\/sub>=1, c3<\/sub>=-5.<\/strong><\/p>\n Consider a3<\/sub> [b1<\/sub>c2<\/sub>-b2<\/sub>c1<\/sub>] + b3 <\/sub>[c1<\/sub>a2<\/sub>-c2<\/sub>a1<\/sub>] + c3 <\/sub>[a1<\/sub>b2<\/sub>-a2<\/sub>b1<\/sub>]<\/strong><\/p>\n = 3 (3 \u2013 8) + 1 (-8-2) + (-5) (-2-3)<\/strong><\/p>\n = -15 -10 + 25<\/strong><\/p>\n =0<\/strong><\/p>\n Since the condition for concurrency is satisfied, the given lines are concurrent.<\/p>\n Still need help with Mathematics? Please read more about our Mathematics tutoring services<\/a>.<\/p>\n SchoolTutoring Academy<\/span> is the premier educational services company for K-12 and college students. We offer tutoring programs for students in K-12, AP classes, and college. To learn more about how we help parents and students in Parksville visit: Tutoring in Parksville . <\/a><\/p>\n","protected":false},"excerpt":{"rendered":" The point of intersection of two or more lines is a point which lies on all the given lies. It means the equations of all the given lines must be satisfied by the intersection point. This point of intersection of lines is called the \u201cpoint of concurrency\u201d.\u00a0 Finding this point of concurrency of two lines […]<\/p>\n","protected":false},"author":19,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_acf_changed":false,"inline_featured_image":false,"footnotes":""},"categories":[4,10],"tags":[538,846,849,2153,2200,2556],"class_list":["post-3289","post","type-post","status-publish","format-standard","hentry","category-calculus","category-geometry","tag-do-lines-on-graphs-cross","tag-how-do-you-calculate-on-graphs","tag-how-do-you-draw-lines","tag-what-does-concurrency-mean","tag-what-is-an-intersection","tag-when-do-lines-meet"],"acf":[],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3289","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/users\/19"}],"replies":[{"embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/comments?post=3289"}],"version-history":[{"count":0,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/posts\/3289\/revisions"}],"wp:attachment":[{"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/media?parent=3289"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/categories?post=3289"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/schooltutoring.com\/help\/wp-json\/wp\/v2\/tags?post=3289"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Condition for point of concurrency of 3 lines:<\/h4>\n