{"id":195973,"date":"2025-03-01T17:09:25","date_gmt":"2025-03-01T17:09:25","guid":{"rendered":"https:\/\/learnexams.com\/blog\/?p=195973"},"modified":"2025-03-01T17:09:28","modified_gmt":"2025-03-01T17:09:28","slug":"a-line-has-slope-3-and-y-intercept-5","status":"publish","type":"post","link":"https:\/\/www.learnexams.com\/blog\/2025\/03\/01\/a-line-has-slope-3-and-y-intercept-5\/","title":{"rendered":"A line has slope-3 and y-intercept 5"},"content":{"rendered":"\n

A line has slope-3 and y-intercept 5. Find a vector equation of the line. <\/p>\n\n\n\n

a. [x, y] = [0, 5] + [1.-3] <\/p>\n\n\n\n

b. IX v1 = [5,0] + 110.-31<\/p>\n\n\n\n

c. LX. y = [1, -3] + [0, ] <\/p>\n\n\n\n

d. x. v = (-3,5] + 1-3, -31<\/p>\n\n\n\n

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

To find the vector equation of a line with a given slope and y-intercept, let’s first break down the problem and understand the components involved.<\/p>\n\n\n\n

Step-by-Step Solution:<\/h3>\n\n\n\n

We are given the following information:<\/p>\n\n\n\n

    \n
  1. Slope of the line (m)<\/strong> = -3<\/li>\n\n\n\n
  2. Y-intercept<\/strong> = 5<\/li>\n<\/ol>\n\n\n\n

    The equation of a line in slope-intercept form is given as:
    [
    y = mx + b
    ]
    where:<\/p>\n\n\n\n

      \n
    • ( m ) is the slope,<\/li>\n\n\n\n
    • ( b ) is the y-intercept.<\/li>\n<\/ul>\n\n\n\n

      Substituting the given values:
      [
      y = -3x + 5
      ]
      This is the equation of the line in terms of (x) and (y).<\/p>\n\n\n\n

      Now, let’s express this line in vector form. A vector equation of a line can be written as:
      [
      [x, y] = [x_0, y_0] + t \\cdot [a, b]
      ]
      where:<\/p>\n\n\n\n

        \n
      • ([x_0, y_0]) is a point on the line (often the y-intercept or any known point),<\/li>\n\n\n\n
      • ([a, b]) is a direction vector representing the slope (the change in (x) and (y)),<\/li>\n\n\n\n
      • (t) is a scalar parameter that scales the direction vector.<\/li>\n<\/ul>\n\n\n\n

        Step 1: Identify a point on the line<\/h3>\n\n\n\n

        We know that the y-intercept is 5, so one point on the line is:
        [
        [x_0, y_0] = [0, 5]
        ]
        This is the point where the line crosses the y-axis.<\/p>\n\n\n\n

        Step 2: Direction vector based on the slope<\/h3>\n\n\n\n

        The slope of the line is -3. This means for every 1 unit increase in (x), (y) decreases by 3 units. Therefore, the direction vector, ([a, b]), corresponding to the slope is:
        [
        [a, b] = [1, -3]
        ]
        Here, for each unit increase in (x), (y) changes by -3.<\/p>\n\n\n\n

        Step 3: Form the vector equation<\/h3>\n\n\n\n

        Now that we have both the point ([0, 5]) and the direction vector ([1, -3]), we can write the vector equation of the line as:
        [
        [x, y] = [0, 5] + t \\cdot [1, -3]
        ]
        This is the vector equation of the line, where (t) is a scalar that can take any real value.<\/p>\n\n\n\n

        Correct Answer:<\/h3>\n\n\n\n

        From the options provided, the correct vector equation is:<\/p>\n\n\n\n

        a.<\/strong> ([x, y] = [0, 5] + t \\cdot [1, -3])<\/p>\n\n\n\n

        This is the equation of the line in vector form.<\/p>\n\n\n\n

        Explanation:<\/h3>\n\n\n\n
          \n
        • The point ([0, 5]) is a known point on the line, which is the y-intercept.<\/li>\n\n\n\n
        • The direction vector ([1, -3]) is derived from the slope of -3, indicating that for each step in the positive (x)-direction, the value of (y) decreases by 3.<\/li>\n<\/ul>\n\n\n\n

          Thus, this vector equation correctly describes the line with slope -3 and y-intercept 5.<\/p>\n\n\n\n

          I will now generate a visual representation of this line for you.<\/p>\n\n\n\n

          \"\"<\/figure>\n\n\n\n

          Here is the visual representation of the line with slope -3 and y-intercept 5. The line passes through the point (0, 5) on the y-axis, and its slope causes it to slope downward as it moves along the x-axis. This image shows how the line behaves based on the vector equation we’ve derived.<\/p>\n","protected":false},"excerpt":{"rendered":"

          A line has slope-3 and y-intercept 5. Find a vector equation of the line. a. [x, y] = [0, 5] + [1.-3] b. IX v1 = [5,0] + 110.-31 c. LX. y = [1, -3] + [0, ] d. x. v = (-3,5] + 1-3, -31 The Correct Answer and Explanation is : To find […]<\/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-195973","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195973","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=195973"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/195973\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=195973"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=195973"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=195973"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}