Solve by using Cramer’s rule the equations:<\/p>\n\n\n\n
6x – 7y = 16
9x – 5y = 35<\/p>\n\n\n\n
The correct answer and explanation is :<\/strong><\/mark><\/p>\n\n\n\n To solve the system of equations using Cramer’s Rule<\/strong>, we follow these steps:<\/p>\n\n\n\n The system can be written as: A\u22c5X=BA \\cdot X = B<\/p>\n\n\n\n where The determinant of matrix AA, denoted as det(A)<\/strong>, is calculated as: det(A)=(6)(\u22125)\u2212(\u22127)(9)\\text{det}(A) = (6)(-5) – (-7)(9) =\u221230+63=33= -30 + 63 = 33<\/p>\n\n\n\n Since det(A) \u2260 0<\/strong>, the system has a unique solution.<\/p>\n\n\n\n To find xx, replace the first column of AA with BB: Ax=[16\u2212735\u22125]A_x = \\begin{bmatrix} 16 & -7 \\\\ 35 & -5 \\end{bmatrix}<\/p>\n\n\n\n The determinant of AxA_x: det(Ax)=(16)(\u22125)\u2212(\u22127)(35)\\text{det}(A_x) = (16)(-5) – (-7)(35) =\u221280+245=165= -80 + 245 = 165<\/p>\n\n\n\n Now, using Cramer\u2019s Rule: x=det(Ax)det(A)=16533=5x = \\frac{\\text{det}(A_x)}{\\text{det}(A)} = \\frac{165}{33} = 5<\/p>\n\n\n\n To find yy, replace the second column of AA with BB: Ay=[616935]A_y = \\begin{bmatrix} 6 & 16 \\\\ 9 & 35 \\end{bmatrix}<\/p>\n\n\n\n The determinant of AyA_y: det(Ay)=(6)(35)\u2212(16)(9)\\text{det}(A_y) = (6)(35) – (16)(9) =210\u2212144=66= 210 – 144 = 66<\/p>\n\n\n\n Now, using Cramer\u2019s Rule: y=det(Ay)det(A)=6633=2y = \\frac{\\text{det}(A_y)}{\\text{det}(A)} = \\frac{66}{33} = 2<\/p>\n\n\n\n x=5,y=2x = 5, \\quad y = 2<\/p>\n\n\n\n Cramer\u2019s Rule is a method for solving a system of linear equations using determinants. It is applicable when the coefficient matrix has a nonzero determinant. This method is especially useful for small systems because it provides an explicit formula for the variables.<\/p>\n\n\n\n The given system consists of two linear equations with two variables, xx and yy. We first express the system in matrix form, where the coefficient matrix AA contains the coefficients of xx and yy, and the constant matrix BB contains the constants from the right side of the equations.<\/p>\n\n\n\n The determinant of matrix AA is computed using the formula: det(A)=(a1b2\u2212a2b1)\\text{det}(A) = (a_1b_2 – a_2b_1)<\/p>\n\n\n\n where a1,b1a_1, b_1 are elements of the first row and a2,b2a_2, b_2 are elements of the second row. If det(A)\u22600\\text{det}(A) \\neq 0, then unique solutions exist.<\/p>\n\n\n\n Next, we find determinants AxA_x and AyA_y by replacing respective columns of AA with BB and computing their determinants. Finally, Cramer\u2019s Rule states: x=det(Ax)det(A),y=det(Ay)det(A)x = \\frac{\\text{det}(A_x)}{\\text{det}(A)}, \\quad y = \\frac{\\text{det}(A_y)}{\\text{det}(A)}<\/p>\n\n\n\n By substituting the computed values, we get x=5x = 5 and y=2y = 2. This method provides an exact and efficient solution when determinants are easily computable.<\/p>\n","protected":false},"excerpt":{"rendered":" Solve by using Cramer’s rule the equations: 6x – 7y = 169x – 5y = 35 The correct answer and explanation is : To solve the system of equations using Cramer’s Rule, we follow these steps: Given System of Equations: Step 1: Write the system in matrix form The system can be written as: A\u22c5X=BA […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","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":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"footnotes":""},"categories":[25],"tags":[],"class_list":["post-203353","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203353","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/comments?post=203353"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/203353\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=203353"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=203353"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=203353"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Given System of Equations:<\/h3>\n\n\n\n
\n
Step 1: Write the system in matrix form<\/h3>\n\n\n\n
A=[6\u221279\u22125]A = \\begin{bmatrix} 6 & -7 \\\\ 9 & -5 \\end{bmatrix},
X=[xy]X = \\begin{bmatrix} x \\\\ y \\end{bmatrix},
B=[1635]B = \\begin{bmatrix} 16 \\\\ 35 \\end{bmatrix}.<\/p>\n\n\n\nStep 2: Find the determinant of matrix AA<\/h3>\n\n\n\n
Step 3: Find xx and yy<\/h3>\n\n\n\n
Finding xx<\/h4>\n\n\n\n
Finding yy<\/h4>\n\n\n\n
Final Answer:<\/h3>\n\n\n\n
Explanation (300 words)<\/h3>\n\n\n\n