2.3-bo'lim : Chiziqli tenglamalarni qo'llash
6. Jon 28 soat ichida uyni bo'yashga qodir. Jon va Deyv birgalikda 17 soat ichida uyni bo'yashlari mumkin. Deyv uyni o'zi bo'yash uchun qancha vaqt kerak bo'ladi?
Yechim
Shunday qilib, agar biz uyni bo'yashni bitta ish deb hisoblasak, agar Jon va Deyv uyni bo'yash uchun birgalikda ishlasa, quyidagi so'z tenglamasiga ega bo'lamiz.
\[\left( \begin{array}{c}{\mbox{Portion of job }}\\ {\mbox{ done by John}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Portion of job}}\\ {\mbox{done by Dave}}\end{array} \right) = 1{\mbox{ Job}}\]Biz bilamizki, Ish qismi = Ish tezligi X ish vaqti, shuning uchun bu quyidagi so'z tenglamasini beradi.
\[\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of John}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of John}}\end{array} \right) + \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of Dave}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of Dave}}\end{array} \right) = 1\]2-qadam
Jon 28 soat ichida uyni bo'yashini bilamiz, shuning uchun biz Jonning ish tezligini aniqlash uchun quyidagi tenglamadan foydalanishimiz mumkin.
\[\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of John}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of John}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of John}}\end{array} \right)\left( {28} \right) = 1\hspace{0.25in} \Rightarrow \hspace{0.25in}{\mbox{Work Rate of John = }}\frac{1}{{28}}\]Xuddi shunday, agar biz \(t\) ni Deyvning uyni o'zi bo'yash uchun qancha vaqt sarflaganiga ruxsat bersak, Deyvning vaqti va ish tezligi o'rtasida quyidagi bog'liqlik bo'ladi.
\[\left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of Dave}}\end{array} \right)\left( \begin{array}{c}{\mbox{Work Time}}\\ {\mbox{ of Dave}}\end{array} \right) = \left( \begin{array}{c}{\mbox{Work Rate}}\\ {\mbox{ of Dave}}\end{array} \right)\left( t \right) = 1 \hspace{0.25in} \Rightarrow \hspace{0.25in}{\mbox{Work Rate of Dave = }}\frac{1}{t}\]3-qadam
Endi biz ikkinchi bosqichdagi ma'lumotlarni kiritishimiz mumkin, shuningdek, Jon va Deyvga uyni birinchi qadamdan boshlab so'z tenglamasiga bo'yash uchun 17 soat vaqt ketadi,
\[\begin{align*}\left( {\frac{1}{{28}}} \right)\left( {17} \right) + \left( {\frac{1}{t}} \right)\left( {17} \right) & = 1\\ \frac{{17}}{{28}} + \frac{{17}}{t} & = 1\end{align*}\]4-qadam
Endi biz buni \(t\) uchun hal qila olamiz.
\[\begin{align*}\frac{{17}}{t} & = 1 - \frac{{17}}{{28}}\\ \frac{{17}}{t} & = \frac{{11}}{{28}}\\ 17 & = \frac{{11}}{{28}}t\\ \frac{{476}}{{11}} & = t\hspace{0.25in} \Rightarrow \hspace{0.25in}t = 43.2727\end{align*}\]Demak, Deyv uyni o'zi bo'yash uchun taxminan 43,2727 soat vaqt ketadi.