{"id":5066,"date":"2025-05-03T11:46:00","date_gmt":"2025-05-03T15:46:00","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=5066"},"modified":"2025-10-06T10:56:34","modified_gmt":"2025-10-06T14:56:34","slug":"algebra-challenge-5-3-2025","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=5066","title":{"rendered":"Algebra Challenge – 2025\/05\/03"},"content":{"rendered":"\n

Let $n$ be the real number. Define $[n]$ be the integer part of $n$, and $\\{n\\}$ be the decimal part of $n$. Solve the following equations:<\/p>\n\n\n\n$$\\begin{array}{ccccccc}\n\\{x\\} & + & [y] & + & \\{z\\} & = 2.9\\\\\n\\{y\\} & + & [z] & + & \\{x\\} & = 5.3\\\\\n\\{z\\} & + & [x] & + & \\{y\\} & = 4.0\\\\\n\\end{array}$$\n\n\n\n

Click here<\/a> for the solution.<\/p>\n

\n\n\n\n

Solution<\/strong>: Because $\\{z\\}+[x]+\\{y\\}=4.0$, $\\{z\\}+\\{y\\}$ is either $0.0$ or $1.0$. <\/p>\n\n\n\n

If $\\{z\\}+\\{y\\}=0.0$, then $\\{y\\}=\\{z\\}=0.0$. This would result in $\\{x\\}=0.9$ and $\\{x\\}=0.3$, which is impossible for ${x}$ to take two values at the same time. Therefore $\\{z\\}+\\{y\\}=1.0$.<\/p>\n\n\n\n

Because $\\{x\\}+[y]+\\{z\\}=2.9$, we have $\\{x\\}+\\{z\\}=0.9$ or $\\{x\\}+\\{z\\}=1.9$<\/p>\n\n\n\n

If $\\{x\\}+\\{z\\}=1.9$, therefore $\\{x\\}>0.9$, $\\{z\\}>0.9$. Therefore $\\{y\\}+\\{x\\}=1.3$. We have:<\/p>\n\n\n\n$$\\begin{array}{ccccccc}\n\\{x\\} & + & \\{z\\} & = 1.9\\\\\n\\{y\\} & + & \\{x\\} & = 1.3\\\\\n\\{z\\} & + & \\{y\\} & = 1.0\\\\\n\\end{array}$$\n\n\n\n

Solving the above equations, we have $\\{x\\}=1.1$, $\\{y\\}=0.2$, and $\\{z\\}=0.8$, which is invalid solution as $\\{x\\}\\ge 1.0$. Therefore $\\{x\\}+\\{z\\}=0.9$.<\/p>\n\n\n\n

Therefore, $\\{x\\}+\\{z\\}$ is either $0.3$ or $1.3$.<\/p>\n\n\n\n

If $\\{x\\}+\\{z\\}=0.3$, we have<\/p>\n\n\n\n$$\\begin{array}{ccccccc}\n\\{x\\} & + & \\{z\\} & = 0.9\\\\\n\\{y\\} & + & \\{x\\} & = 0.3\\\\\n\\{z\\} & + & \\{y\\} & = 1.0\\\\\n\\end{array}$$\n\n\n\n

Solving the above equations, we have $\\{x\\}=0.1$, $\\{y\\}=0.2$, $\\{z\\}=0.8$. This results in $x=3.1$, $y=2.2$, and $z=5.8$.<\/p>\n\n\n\n

If $\\{x\\}+\\{z\\}=1.3$, we have<\/p>\n\n\n\n$$\\begin{array}{ccccccc}\n\\{x\\} & + & \\{z\\} & = 0.9\\\\\n\\{y\\} & + & \\{x\\} & = 1.3\\\\\n\\{z\\} & + & \\{y\\} & = 1.0\\\\\n\\end{array}$$\n\n\n\n

Solving the above equations, we have $\\{x\\}=0.6$, $\\{y\\}=0.7$, $\\{z\\}=0.3$. This results in $x=3.6$, $y=2.7$, and $z=4.3$.<\/p>\n\n\n\n

In conclusion there are two solutions for $(x, y, z)$, i.e. $(3.1, 2.2, 5.8)$ and $(3.6, 2.7, 4.3)$.<\/p>\n\n\n\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Let $n$ be the real number. Define $[n]$ be the integer part of $n$, and $\\{n\\}$ be the decimal part of $n$. Solve the following equations: $$\\begin{array}{ccccccc} \\{x\\} & + & [y] & + & \\{z\\} & = 2.9\\\\ \\{y\\} … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[13],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5066"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=5066"}],"version-history":[{"count":27,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5066\/revisions"}],"predecessor-version":[{"id":5344,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/5066\/revisions\/5344"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=5066"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=5066"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=5066"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}