Q.1) Chase and Sara went to the candy store. Chase bought 2 pieces of fudge and 1 piece of bubble gum for a total of $32. Sara bought 1 piece of fudge and 3 pieces of bubble gum for a total of $36. What are the prices of each piece of fudge and piece of bubble gum?
Q.2) Solve the following systems of Linear equations using Gauss-Jordan method (show all the steps\/part a is a bonus question) a) x+2y-z=1 2x+y+4z=2 3x+3y+4z=1<\/p>\n\n\n\n
Q.3) Solve the following system using the Inverse matrix 5x+y=4 2x-3y=5<\/p>\n\n\n\n
The correct answer and explanation is:<\/mark><\/strong><\/p>\n\n\n\n Let the price of a piece of fudge be FF and the price of a piece of bubble gum be BB. The given equations are: 2F+B=322F + B = 32 F+3B=36F + 3B = 36<\/p>\n\n\n\n The system of equations is:<\/p>\n\n\n\n [12\u22121\u22231214\u22232334\u22231]\\begin{bmatrix} 1 & 2 & -1 & | & 1 \\\\ 2 & 1 & 4 & | & 2 \\\\ 3 & 3 & 4 & | & 1 \\end{bmatrix}<\/p>\n\n\n\n Final matrix: [100\u22233010\u2223\u22122001\u2223\u22122]\\begin{bmatrix} 1 & 0 & 0 & | & 3 \\\\ 0 & 1 & 0 & | & -2 \\\\ 0 & 0 & 1 & | & -2 \\end{bmatrix}<\/p>\n\n\n\n x=3,\u2009y=\u22122,\u2009z=\u22122x = 3, \\, y = -2, \\, z = -2<\/p>\n\n\n\n The system of equations is:<\/p>\n\n\n\n A=[512\u22123],X=[xy],B=[45]\\mathbf{A} = \\begin{bmatrix} 5 & 1 \\\\ 2 & -3 \\end{bmatrix}, \\quad \\mathbf{X} = \\begin{bmatrix} x \\\\ y \\end{bmatrix}, \\quad \\mathbf{B} = \\begin{bmatrix} 4 \\\\ 5 \\end{bmatrix} AX=B\\mathbf{AX} = \\mathbf{B}<\/p>\n\n\n\n A\u22121=1det(A)[\u22123\u22121\u221225]\\mathbf{A}^{-1} = \\frac{1}{\\text{det}(\\mathbf{A})} \\begin{bmatrix} -3 & -1 \\\\ -2 & 5 \\end{bmatrix} det(A)=(5)(\u22123)\u2212(2)(1)=\u221215\u22122=\u221217\\text{det}(\\mathbf{A}) = (5)(-3) – (2)(1) = -15 – 2 = -17 A\u22121=1\u221217[\u22123\u22121\u221225]=[317117217\u2212517]\\mathbf{A}^{-1} = \\frac{1}{-17} \\begin{bmatrix} -3 & -1 \\\\ -2 & 5 \\end{bmatrix} = \\begin{bmatrix} \\frac{3}{17} & \\frac{1}{17} \\\\ \\frac{2}{17} & -\\frac{5}{17} \\end{bmatrix}<\/p>\n\n\n\n X=A\u22121B\\mathbf{X} = \\mathbf{A}^{-1} \\mathbf{B} X=[317117217\u2212517][45]\\mathbf{X} = \\begin{bmatrix} \\frac{3}{17} & \\frac{1}{17} \\\\ \\frac{2}{17} & -\\frac{5}{17} \\end{bmatrix} \\begin{bmatrix} 4 \\\\ 5 \\end{bmatrix} X=[3(4)+1(5)172(4)\u22125(5)17]\\mathbf{X} = \\begin{bmatrix} \\frac{3(4) + 1(5)}{17} \\\\ \\frac{2(4) – 5(5)}{17} \\end{bmatrix} X=[12+5178\u22122517]\\mathbf{X} = \\begin{bmatrix} \\frac{12 + 5}{17} \\\\ \\frac{8 – 25}{17} \\end{bmatrix} X=[1717\u22121717]\\mathbf{X} = \\begin{bmatrix} \\frac{17}{17} \\\\ \\frac{-17}{17} \\end{bmatrix} X=[1\u22121]\\mathbf{X} = \\begin{bmatrix} 1 \\\\ -1 \\end{bmatrix}<\/p>\n\n\n\n x=1,\u2009y=\u22121x = 1, \\, y = -1<\/p>\n\n\n\n Explanation<\/strong>: Using the inverse matrix method allows for a systematic approach to solve linear systems. By finding the matrix inverse, the system is transformed into a simple matrix multiplication problem. This method is especially useful when solving systems programmatically or with larger datasets, as it avoids manual row operations.<\/p>\n","protected":false},"excerpt":{"rendered":" Q.1) Chase and Sara went to the candy store. Chase bought 2 pieces of fudge and 1 piece of bubble gum for a total of $32. Sara bought 1 piece of fudge and 3 pieces of bubble gum for a total of $36. What are the prices of each piece of fudge and piece of […]<\/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-182765","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182765","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=182765"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/182765\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=182765"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=182765"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=182765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}Solution to Q.1<\/strong><\/h3>\n\n\n\n
Step 1: Solve the system of equations.<\/h4>\n\n\n\n
\n
Final Answer:<\/h4>\n\n\n\n
\n
\n\n\n\nSolution to Q.2<\/strong> (Gauss-Jordan Method)<\/h3>\n\n\n\n
\n
Step 1: Write the augmented matrix<\/h4>\n\n\n\n
Step 2: Perform row operations to reach reduced row-echelon form.<\/h4>\n\n\n\n
\n
\n
Final Answer:<\/h4>\n\n\n\n
\n\n\n\nSolution to Q.3<\/strong> (Using Inverse Matrix)<\/h3>\n\n\n\n
\n
Step 1: Write in matrix form.<\/h4>\n\n\n\n
Step 2: Find the inverse of A\\mathbf{A}.<\/h4>\n\n\n\n
Step 3: Solve for X\\mathbf{X}.<\/h4>\n\n\n\n
Final Answer:<\/h4>\n\n\n\n
\n\n\n\n