{"id":232236,"date":"2025-06-11T21:41:09","date_gmt":"2025-06-11T21:41:09","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=232236"},"modified":"2025-06-11T21:41:12","modified_gmt":"2025-06-11T21:41:12","slug":"determine-if-the-system-has-a-nontrivial-solution","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/06\/11\/determine-if-the-system-has-a-nontrivial-solution\/","title":{"rendered":"Determine if the system has a nontrivial solution."},"content":{"rendered":"\n

Determine if the system has a nontrivial solution. Try to use as few row operations as possible. 6x^2 + 7xz 6x^3 + 4x^3 Choose the correct answer below: The system has only a trivial solution: The system has a nontrivial solution: It is impossible to determine.<\/p>\n\n\n\n

The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n

ChatGPT said:<\/h6>\n\n\n\n

To determine whether a system has a nontrivial solution<\/strong>, we need to understand what the system is. However, your question seems to mix a few different expressions:
\u201c6x\u00b2 + 7xz 6x\u00b3 + 4x\u00b3<\/strong>\u201d \u2014 this looks more like a list of algebraic expressions than a system of equations. Let’s interpret it carefully.<\/p>\n\n\n\n

\n\n\n\n

Step 1: Clarify the Problem<\/h3>\n\n\n\n

You’re likely being asked to determine if a system of equations<\/strong> (possibly homogeneous) has a nontrivial solution<\/strong>. A homogeneous system has the form:Ax\u20d7=0\u20d7A \\vec{x} = \\vec{0}Ax=0<\/p>\n\n\n\n

and has a nontrivial solution<\/strong> (i.e., a solution other than the zero vector) if and only if<\/strong> the system has free variables<\/strong>, or equivalently, if the determinant of the coefficient matrix is zero<\/strong> (i.e., the matrix is not full rank).<\/p>\n\n\n\n

But from what you typed:<\/p>\n\n\n\n

    \n
  • \u201c6x\u00b2 + 7xz 6x\u00b3 + 4x\u00b3<\/strong>\u201d could be meant to represent the expressions in equations, or possibly a vector form.<\/li>\n<\/ul>\n\n\n\n

    Let\u2019s assume this is a homogeneous system where the coefficients or expressions might represent rows of a matrix.<\/p>\n\n\n\n

    Suppose we are given a vector space expression like:v\u20d71=6×2+7xz,v\u20d72=6×3+4×3=10×3\\vec{v}_1 = 6x^2 + 7xz, \\quad \\vec{v}_2 = 6x^3 + 4x^3 = 10x^3v1\u200b=6×2+7xz,v2\u200b=6×3+4×3=10×3<\/p>\n\n\n\n

    If these vectors are viewed as linear combinations or parts of a homogeneous system, we ask:<\/p>\n\n\n\n

    Do these expressions (vectors) form a linearly independent set?<\/strong><\/p>\n\n\n\n

    Let\u2019s consider the following:<\/p>\n\n\n\n

      \n
    • The expression 6×2+7xz6x^2 + 7xz6x2+7xz involves degree-2 monomials<\/strong>: x2x^2×2, xzxzxz<\/li>\n\n\n\n
    • The expression 10x310x^310×3 involves only the degree-3 monomial<\/strong> x3x^3×3<\/li>\n<\/ul>\n\n\n\n

      Since these expressions involve different degrees<\/strong> and distinct monomials<\/strong>, they are linearly independent<\/strong>.<\/p>\n\n\n\n

      Thus, if these are rows or columns of a matrix representing a homogeneous system, and if the number of unknowns exceeds the number of independent equations, there would be a nontrivial solution<\/strong>.<\/p>\n\n\n\n

      \n\n\n\n

      \u2705 Final Answer:<\/h3>\n\n\n\n

      The system has a nontrivial solution.<\/strong><\/p>\n\n\n\n

      \n\n\n\n

      Explanation <\/h3>\n\n\n\n

      A homogeneous system<\/strong> of equations is one in which all equations are set equal to zero, such as:a1x+a2y+a3z=0a_1x + a_2y + a_3z = 0a1\u200bx+a2\u200by+a3\u200bz=0<\/p>\n\n\n\n

      A trivial solution<\/strong> occurs when all variables are zero (e.g., x=y=z=0x = y = z = 0x=y=z=0). A nontrivial solution<\/strong> exists if there are nonzero values<\/strong> of the variables that also satisfy the system.<\/p>\n\n\n\n

      To determine whether a nontrivial solution exists, we analyze the linear independence<\/strong> of the expressions provided.<\/p>\n\n\n\n

      Given the expressions:6×2+7xz,6×3+4×3=10x36x^2 + 7xz,\\quad 6x^3 + 4x^3 = 10x^36×2+7xz,6×3+4×3=10×3<\/p>\n\n\n\n

      These can be seen as vectors in a polynomial space. The first expression, 6×2+7xz6x^2 + 7xz6x2+7xz, contains monomials x2x^2×2 and xzxzxz, both of degree 2<\/strong>. The second expression, 10x310x^310×3, contains only x3x^3×3, which is of degree 3<\/strong>.<\/p>\n\n\n\n

      Because these expressions involve distinct monomials<\/strong> and are of different degrees<\/strong>, they cannot be linear combinations of one another. Therefore, they are linearly independent<\/strong>.<\/p>\n\n\n\n

      If such expressions form part of a matrix in a homogeneous linear system, then each independent vector adds to the rank<\/strong> of the matrix. A system has a nontrivial solution if the number of variables exceeds<\/strong> the rank (i.e., the number of linearly independent equations). Since these expressions do not overlap in their monomial components, and assuming there are at least 3 variables (e.g., x,y,zx, y, zx,y,z), we would have more variables than independent equations.<\/p>\n\n\n\n

      Thus, the system allows for at least one free variable<\/strong>, which implies the existence of nontrivial solutions<\/strong>.<\/p>\n\n\n\n

      Hence, the correct conclusion is:<\/p>\n\n\n\n

      \n

      \u2705 The system has a nontrivial solution<\/p>\n\n\n\n

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

      \"\"<\/figure>\n","protected":false},"excerpt":{"rendered":"

      Determine if the system has a nontrivial solution. Try to use as few row operations as possible. 6x^2 + 7xz 6x^3 + 4x^3 Choose the correct answer below: The system has only a trivial solution: The system has a nontrivial solution: It is impossible to determine. The Correct Answer and Explanation is: ChatGPT said: To […]<\/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-232236","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232236","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=232236"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/232236\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=232236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=232236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=232236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}