Question 1136263
.


            Really,  such a problem can stump unexperienced student.


            But it is not so complicated.


            The solution is as follows.



<pre>
From equation  8x + 2y = 18, express  y = {{{(18-8x)/2}}} = -4x + 9  and substitute it into the expression  


    f(x,y) = {{{x^2 + y^2}}} = {{{x^2 + (-4x+9)^2}}} = {{{x^2 + 16x^2 - 72x + 81}}} = {{{17x^2 - 72x + 81}}}.     (1)


Now your function is presented as a quadratic function of ONE argument.


You can easily find its minimum by using this very well known shortcut   {{{x[min]}}} = " {{{-b/(2a)}}} ".


In this case  a = 17 and b = -72,  so  {{{x[min]}}} = {{{-(-72)/(2*17)}}} = {{{72/34}}}.


So, you just know the value of " x " where the function has the minimum.


By knowing  {{{x[min]}}},  you can easily calculate  {{{y[min]}}} = {{{-4*x[min] + 9}}}. 


I leave this simple arithmetic for you to complete it on your own.
</pre>

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On finding the maximum/minimum of a quadratic function see my lessons 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>HOW TO complete the square to find the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-How-to-complete-the-square-of-a-quadratic-function-to-find-its-minimum-maximum.lesson>Briefly on finding the minimum/maximum of a quadratic function</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-complete-the-square-to-find-the-vertex-of-a-quadratic-function.lesson>HOW TO complete the square to find the vertex of a parabola</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Briefly-on-finding-the-vertex-of-a-parabola.lesson>Briefly on finding the vertex of a parabola</A>

in this site.


Also, &nbsp;you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lessons are the part of this textbook under the topic "<U>Finding minimum/maximum of quadratic functions</U>". 



Save the link to this online textbook together with its description


Free of charge online textbook in ALGEBRA-I
https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson


to your archive and use it when it is needed.