For the system of equations x + y = 1 and 4x + 5y = 7, find x and y so that you can determine 4xy.<\/p>\n\n\n\n
The correct answer and explanation is:<\/strong><\/mark><\/p>\n\n\n\n Let’s solve the system of equations: {x+y=14x+5y=7\\begin{cases} x + y = 1 \\\\ 4x + 5y = 7 \\end{cases}<\/p>\n\n\n\n Step 1: Express one variable in terms of the other<\/strong><\/p>\n\n\n\n From the first equation: x+y=1\u2005\u200a\u27f9\u2005\u200ay=1\u2212xx + y = 1 \\implies y = 1 – x<\/p>\n\n\n\n Step 2: Substitute y=1\u2212xy = 1 – x into the second equation<\/strong> 4x+5(1\u2212x)=74x + 5(1 – x) = 7<\/p>\n\n\n\n Simplify: 4x+5\u22125x=74x + 5 – 5x = 7 (4x\u22125x)+5=7(4x – 5x) + 5 = 7 \u2212x+5=7- x + 5 = 7<\/p>\n\n\n\n Step 3: Solve for xx<\/strong> \u2212x=7\u22125- x = 7 – 5 \u2212x=2\u2005\u200a\u27f9\u2005\u200ax=\u22122- x = 2 \\implies x = -2<\/p>\n\n\n\n Step 4: Find yy using y=1\u2212xy = 1 – x<\/strong> y=1\u2212(\u22122)=1+2=3y = 1 – (-2) = 1 + 2 = 3<\/p>\n\n\n\n Step 5: Calculate 4xy4xy<\/strong> 4xy=4\u00d7(\u22122)\u00d73=4\u00d7(\u22126)=\u2212244xy = 4 \\times (-2) \\times 3 = 4 \\times (-6) = -24<\/p>\n\n\n\n We started by isolating one variable, yy, in the first equation. This is a common technique for solving systems of linear equations. Substituting yy into the second equation allows us to solve for xx in terms of constants only.<\/p>\n\n\n\n After substitution, the second equation becomes a single-variable linear equation which is straightforward to solve. We find x=\u22122x = -2, then back-substitute to find y=3y = 3.<\/p>\n\n\n\n Finally, the problem asks to find 4xy4xy, not just xx and yy. Multiplying xx and yy and then by 4 yields \u221224-24.<\/p>\n\n\n\n This approach is simple and effective for any system of two linear equations. It ensures a clear step-by-step solution that avoids errors and helps understand the relations between variables.<\/p>\n","protected":false},"excerpt":{"rendered":" For the system of equations x + y = 1 and 4x + 5y = 7, find x and y so that you can determine 4xy. The correct answer and explanation is: Let’s solve the system of equations: {x+y=14x+5y=7\\begin{cases} x + y = 1 \\\\ 4x + 5y = 7 \\end{cases} Step 1: Express one […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1],"tags":[],"class_list":["post-23399","post","type-post","status-publish","format-standard","hentry","category-quiz-questions"],"_links":{"self":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23399","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/comments?post=23399"}],"version-history":[{"count":1,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23399\/revisions"}],"predecessor-version":[{"id":23402,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/posts\/23399\/revisions\/23402"}],"wp:attachment":[{"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/media?parent=23399"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/categories?post=23399"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/gaviki.com\/blog\/wp-json\/wp\/v2\/tags?post=23399"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\n\n\nExplanation:<\/h3>\n\n\n\n