Three students are working to find the solution set of this system of equations:
3y = 6x + 6
y = 2x + 2
Use the drop-down menus to complete the statements about each of their methods.<\/p>\n\n\n\n
Pedro Dropdown 1: Intersect, Do not Intersect, Are the same line.<\/p>\n\n\n\n
Pedro Dropdown 2: One, Zero, Infinitely many<\/p>\n\n\n\n
Amy Dropdown: is, is not<\/p>\n\n\n\n
Matt Dropdown 1: Sometimes, Never, Always<\/p>\n\n\n\n
Matt Dropdown 2: One, Zero, Infinitely Many<\/p>\n\n\n\n
The Correct Answer and Explanation is:<\/mark><\/strong><\/p>\n\n\n\n To solve the system of equations and complete the statements about each student’s method, we begin by analyzing the given equations:<\/p>\n\n\n\n We can rewrite the first equation in slope-intercept form to facilitate easier comparison with the second equation.<\/p>\n\n\n\n The first equation is: Both equations represent the same line, as they have identical slopes and y-intercepts. This means the two lines overlap completely.<\/p>\n\n\n\n Pedro’s Statement 1:<\/strong> Pedro Dropdown 1:<\/strong> Are the same line<\/em><\/p>\n\n\n\n Pedro’s Statement 2:<\/strong> Pedro Dropdown 2:<\/strong> Infinitely many<\/em><\/p>\n\n\n\n Amy needs to determine whether the system is<\/strong> or is not<\/strong> consistent. A system of equations is consistent if there is at least one solution. Since both equations represent the same line, there are infinitely many solutions. Therefore, the correct choice for Amy is:<\/p>\n\n\n\n Amy Dropdown:<\/strong> is<\/em><\/p>\n\n\n\n Matt is dealing with the general case, where the two lines might sometimes intersect, never intersect, or always be the same.<\/p>\n\n\n\n Matt’s Statement 1:<\/strong> Matt Dropdown 1:<\/strong> Always<\/em><\/p>\n\n\n\n Matt’s Statement 2:<\/strong> Matt Dropdown 2:<\/strong> Infinitely many<\/em><\/p>\n\n\n\n The system of equations represents two lines that have the same slope (2) and the same y-intercept (2). When two lines in a system of linear equations have identical slopes and intercepts, they coincide, meaning they are the same line. Therefore, the system has infinitely many solutions, as every point on the line is a solution to both equations. This is why Pedro, Amy, and Matt all conclude that the lines are the same, the system is consistent, and there are infinitely many solutions.<\/p>\n","protected":false},"excerpt":{"rendered":" Three students are working to find the solution set of this system of equations:3y = 6x + 6y = 2x + 2Use the drop-down menus to complete the statements about each of their methods. Pedro Dropdown 1: Intersect, Do not Intersect, Are the same line. Pedro Dropdown 2: One, Zero, Infinitely many Amy Dropdown: is, […]<\/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-163555","post","type-post","status-publish","format-standard","hentry","category-exams-certification"],"_links":{"self":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/163555","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=163555"}],"version-history":[{"count":0,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/posts\/163555\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/media?parent=163555"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/categories?post=163555"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.learnexams.com\/blog\/wp-json\/wp\/v2\/tags?post=163555"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}\n
Step 1: Rewriting Equation 1 in Slope-Intercept Form<\/h3>\n\n\n\n
[
3y = 6x + 6
]
Divide both sides by 3:
[
y = 2x + 2
]
Now, both equations are:
[
y = 2x + 2 \\quad \\text{and} \\quad y = 2x + 2
]<\/p>\n\n\n\nStep 2: Analyzing the System<\/h3>\n\n\n\n
Step 3: Answering Pedro’s Statements<\/h3>\n\n\n\n
Pedro needs to decide whether the two lines intersect<\/strong> or are the same<\/strong>. Since both equations represent the same line, they do not intersect at just one point but rather overlap entirely. Therefore, the correct choice for Pedro’s first statement is:<\/p>\n\n\n\n
Since the lines are the same, they intersect infinitely. Therefore, the correct choice for Pedro’s second statement is:<\/p>\n\n\n\nStep 4: Answering Amy’s Statement<\/h3>\n\n\n\n
Step 5: Answering Matt’s Statements<\/h3>\n\n\n\n
Since the lines are the same in this case, they will always<\/strong> coincide, meaning the system will always have solutions. Therefore, the correct choice for Matt’s first statement is:<\/p>\n\n\n\n
Since the two lines overlap, there are infinitely many solutions. Therefore, the correct choice for Matt’s second statement is:<\/p>\n\n\n\nFinal Answers:<\/h3>\n\n\n\n
\n
Explanation:<\/h3>\n\n\n\n