{"id":12612,"date":"2024-03-04T10:13:00","date_gmt":"2024-03-04T04:43:00","guid":{"rendered":"https:\/\/tsboardsolutions.in\/?p=12612"},"modified":"2024-03-05T14:19:45","modified_gmt":"2024-03-05T08:49:45","slug":"ts-10th-class-maths-notes-chapter-4","status":"publish","type":"post","link":"https:\/\/tsboardsolutions.in\/ts-10th-class-maths-notes-chapter-4\/","title":{"rendered":"TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables"},"content":{"rendered":"<p>We are offering <a href=\"https:\/\/tsboardsolutions.in\/ts-10th-class-maths-notes\/\">TS 10th Class Maths Notes<\/a> Chapter 4 Pair of Linear Equations in Two Variables to learn maths more effectively.<\/p>\n<h2>TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables<\/h2>\n<p>\u2192 An Equation of the form ax + by + c = 0, where a, b, c e R and a and b are not both zero, is called a linear equation in two variables x and y.<\/p>\n<p>\u2192 A pair of linear equations in two variables x and y can be represented as follows :<br \/>\na<sub>1<\/sub>x + b<sub>1<\/sub>y + C<sub>1<\/sub> = 0<br \/>\na<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub> = 0<br \/>\nWhere a<sub>1<\/sub>, a<sub>2<\/sub>, b<sub>1<\/sub>, b<sub>2<\/sub>, c<sub>1<\/sub>, c<sub>2<\/sub> are real numbers such that a<sub>1<\/sub><sup>2<\/sup> + b<sub>1<\/sub><sup>2<\/sup> \u2260 0; a<sub>2<\/sub><sup>2<\/sup> + b<sub>2<\/sub><sup>2<\/sup> \u2260 0.<\/p>\n<p>\u2192 A pair of linear Equations in two variables forms a system of simultaneous linear equations.<br \/>\nExample : 3x &#8211; 4y = 2<br \/>\n2x + 5y = 9<\/p>\n<p>\u2192 A pair of values of the variables x and y satisfying each one of the equations that are given is called a solution of the system.<br \/>\nx = 2, y = 1 is a solution of the system of simultaneous linear equations.<br \/>\n3x-4y = 2 &#8230;&#8230;&#8230;&#8230;&#8230; (1)<br \/>\n2x + 5y = 9 &#8230;&#8230;&#8230;&#8230;&#8230; (2)<br \/>\nPutting x = 2 and y = 1 in equation (1), we get<br \/>\nL.H.S. = 3 \u00d7 2 &#8211; 4 \u00d7 1 = 6 &#8211; 4 = 2<br \/>\nR.H.S = 2<br \/>\nL.H.S = R.H.S<br \/>\nSimilarly, put x = 2 and y = 1 in equation (2), we get<br \/>\nL.H.S = 2 \u00d7 2 + 5 \u00d7 1 = 4 + 5 = 9<br \/>\nR.H.S = 9<br \/>\n\u2234 L.H.S = R.H.S<\/p>\n<p>\u2192 A pair of linear equations in two variables can be solved using<\/p>\n<ul>\n<li>Graphical method<\/li>\n<li>Model method<\/li>\n<li>Algebraic method :<br \/>\n(a) Substitution method<br \/>\n(b) Elimination method<br \/>\n(c) Cross-Multiplication method<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/tsboardsolutions.in\/wp-content\/uploads\/2022\/12\/TS-Board-Solutions.png\" alt=\"TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables\" width=\"161\" height=\"15\" \/><\/p>\n<p>\u2192 Graphical method: The graph of a pair of linear equations in two variables is represented by two straight lines.<\/p>\n<ul>\n<li>If the lines intersect at a point, then that point gives the unique solution of the two equations. In this case, the pair of equations is consistent.<\/li>\n<li>If the lines coincide, then there are infinitely many solutions &#8211; each point on the line being a solution. In this case, the pair of equations is dependent (consistent).<\/li>\n<li>If the lines are parallel, then the pair of equations has no solution.<br \/>\nIn this case, the pair of equations is inconsistent.<\/li>\n<\/ul>\n<p>\u2192 The relation that exists between the coefficients and nature of system of equations.<\/p>\n<p>\u2192 If a pair of linear equations is given by a<sub>1<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub> = 0 and a<sub>2<\/sub>x + b<sub>2<\/sub>y + c<sub>2<\/sub> = 0, then<\/p>\n<ul>\n<li>The pair of linear equations is consistent if \\(\\frac{a_1}{a_2} \\neq \\frac{b_1}{b_2}\\)<\/li>\n<li>The pair of linear equation is inconsistent if \\(\\frac{a_1}{a_2}=\\frac{b_1}{b_2} \\neq \\frac{c_1}{c_2}\\)<\/li>\n<li>The pair of linear equation is dependent and consistent if \\(\\frac{a_1}{a_2}=\\frac{b_1}{b_2}=\\frac{c_1}{c_2}\\)<\/li>\n<\/ul>\n<p>Important Formulas:<\/p>\n<ul>\n<li>The pair of linear equations is consistent if \\(\\frac{a_1}{a_2} \\neq \\frac{b_1}{b_2}\\)<\/li>\n<li>The pair of linear equation is inconsistent if \\(\\frac{a_1}{a_2}=\\frac{b_1}{b_2} \\neq \\frac{c_1}{c_2}\\)<\/li>\n<li>The pair of linear equation is dependent and consistent if \\(\\frac{a_1}{a_2}=\\frac{b_1}{b_2}=\\frac{c_1}{c_2}\\)<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/tsboardsolutions.in\/wp-content\/uploads\/2022\/12\/TS-Board-Solutions.png\" alt=\"TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables\" width=\"161\" height=\"15\" \/><\/p>\n<p>Flow Chat:<br \/>\n<img loading=\"lazy\" decoding=\"async\" class=\"alignnone size-full wp-image-12615\" src=\"https:\/\/tsboardsolutions.in\/wp-content\/uploads\/2023\/02\/TS-10th-Class-Maths-Notes-Chapter-4-Pair-of-Linear-Equations-in-Two-Variables-1.png\" alt=\"TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables 1\" width=\"864\" height=\"570\" srcset=\"https:\/\/tsboardsolutions.in\/wp-content\/uploads\/2023\/02\/TS-10th-Class-Maths-Notes-Chapter-4-Pair-of-Linear-Equations-in-Two-Variables-1.png 864w, https:\/\/tsboardsolutions.in\/wp-content\/uploads\/2023\/02\/TS-10th-Class-Maths-Notes-Chapter-4-Pair-of-Linear-Equations-in-Two-Variables-1-300x198.png 300w, https:\/\/tsboardsolutions.in\/wp-content\/uploads\/2023\/02\/TS-10th-Class-Maths-Notes-Chapter-4-Pair-of-Linear-Equations-in-Two-Variables-1-768x507.png 768w\" sizes=\"(max-width: 864px) 100vw, 864px\" \/><\/p>\n<p>William George Horner(1786 &#8211; 1832):<\/p>\n<ul>\n<li>A William George Horner was a British mathematician; he was a schoolmaster; headmaster and schoolkeeper, proficient in classics as well as mathematics.<\/li>\n<li>A He wrote extensively on functional equations, number theory and approximation theory.<\/li>\n<li>A His contribution to approximation theory is honoured in the designation Horner&#8217;s method. The modern invention of the zoetrope, under the name Daedaleum in 1834 has been attributed to him.<\/li>\n<\/ul>\n","protected":false},"excerpt":{"rendered":"<p>We are offering TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables to learn maths more effectively. TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables \u2192 An Equation of the form ax + by + c = 0, where a, b, c e R &#8230; <a title=\"TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables\" class=\"read-more\" href=\"https:\/\/tsboardsolutions.in\/ts-10th-class-maths-notes-chapter-4\/\" aria-label=\"More on TS 10th Class Maths Notes Chapter 4 Pair of Linear Equations in Two Variables\">Read more<\/a><\/p>\n","protected":false},"author":2,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[16],"tags":[],"jetpack_featured_media_url":"","_links":{"self":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/12612"}],"collection":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/users\/2"}],"replies":[{"embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/comments?post=12612"}],"version-history":[{"count":2,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/12612\/revisions"}],"predecessor-version":[{"id":12616,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/posts\/12612\/revisions\/12616"}],"wp:attachment":[{"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/media?parent=12612"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/categories?post=12612"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/tsboardsolutions.in\/wp-json\/wp\/v2\/tags?post=12612"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}