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Can two equations when multiplied sometimes not give graph of both the equations?



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 17/18, 2019 at 00:00UTC (8:00pm US/Eastern)Is this Batman equation for real?Power functions and parabola issueAn equation with graph as a line *segment*: $ sqrt x^2 + (y+12)^2 = 13 - sqrt(x-5)^2 + y^2 $Obtaining constant using the straight line gradient of two equationsTriple Simultaneous Equations not resolving the Equation for a Quadratic FunctionWhy graph of $sin(x+45^circ$) shifts to left instead of rightCan the graph of $tan^-1left(fracsin xxright)$ be expressed as $Ce^-kxcos(omega x + phi)$?How to transform a rational function into a straight line (or viceversa)When can we apply matrix operations to both sides of the equation to solve linear systems?The graph of $sqrt(x+2)^2+y^2 + sqrtx^2 + (y-2)^2 = 6$ is a(n)…Finding the intersection of two 2d vector equations










0












$begingroup$


I wanted graph of this equation:
$$left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right) left(x-yright)=0$$



I want a x=y line as well. So I kept (x-y) on the function side since (x-y)=0 should ultimately give x=y as graph along with graph of other functions. But I don't get the x=y line as a whole. It's not complete and I don't understand why.



I think there's something to do with which other equations I have used. But that doesn't seem to click. Isn't it mathematically correct to multiply two equations with 0 on right sides and get both equations' graph?










share|cite|improve this question











$endgroup$
















    0












    $begingroup$


    I wanted graph of this equation:
    $$left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right) left(x-yright)=0$$



    I want a x=y line as well. So I kept (x-y) on the function side since (x-y)=0 should ultimately give x=y as graph along with graph of other functions. But I don't get the x=y line as a whole. It's not complete and I don't understand why.



    I think there's something to do with which other equations I have used. But that doesn't seem to click. Isn't it mathematically correct to multiply two equations with 0 on right sides and get both equations' graph?










    share|cite|improve this question











    $endgroup$














      0












      0








      0





      $begingroup$


      I wanted graph of this equation:
      $$left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right) left(x-yright)=0$$



      I want a x=y line as well. So I kept (x-y) on the function side since (x-y)=0 should ultimately give x=y as graph along with graph of other functions. But I don't get the x=y line as a whole. It's not complete and I don't understand why.



      I think there's something to do with which other equations I have used. But that doesn't seem to click. Isn't it mathematically correct to multiply two equations with 0 on right sides and get both equations' graph?










      share|cite|improve this question











      $endgroup$




      I wanted graph of this equation:
      $$left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right) left(x-yright)=0$$



      I want a x=y line as well. So I kept (x-y) on the function side since (x-y)=0 should ultimately give x=y as graph along with graph of other functions. But I don't get the x=y line as a whole. It's not complete and I don't understand why.



      I think there's something to do with which other equations I have used. But that doesn't seem to click. Isn't it mathematically correct to multiply two equations with 0 on right sides and get both equations' graph?







      linear-algebra functions graphing-functions






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Mar 26 at 10:30









      David Mitra

      63.8k6102165




      63.8k6102165










      asked Mar 26 at 10:30









      user68153user68153

      113




      113




















          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          You're completely right about "the other equations" being the source of the problem.



          Let me start with something simple. What is the value of
          $$
          x cdot 0 ?
          $$



          It's very tempting to say, "It's $0$, of course, because anything times zero is zero!" But what if $x$ is, for instance, a cheese omelet? Then...well...then multiplication by zero isn't even defined, and you'd have to say "$x cdot 0$ is undefined" rather than "it's zero." You might say that "cheese omelet" isn't really math, but what if we replace $x$ like this:
          $$
          (1/0) cdot 0
          $$

          Then our first factor is undefined, and multiplication by undefined things is undefined, so once again the product is "undefined" rather than zero.



          Now let's look at a more complicated example. I'm going to stick with the real numbers -- no complex numbers allowed! -- and I want to look at
          $$
          sqrtx cdot 0
          $$

          You might say "Well, that's definitely zero!", but what if $x = -3$? Then the square root is undefined, and so the product is undefined as well.




          Here's a theorem that your question hints at:




          Theorem: Suppose $f, g : Bbb R^2 to Bbb R$, and $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
          set of the equation $f(x, y) = 0$. Then the solution set of the
          equation $$ f(x,y) g(x, y) = 0 $$ is exactly $F cup G$.




          In your case, you've got $g(x, y) = x - y$, and the solutions for $x-y = 0$ form a line in the plane; that's the set $G$ in the theorem. You've multiplied it by some complicated mess that I'll temporarily call $f(x, y)$, and looked for solutions to
          $$
          tag1
          f(x, y) cdot g(x, y) = 0
          $$

          (perhaps using Desmos or some other graphing tool), and you're wondering why the solution set of this doesn't include all of $G$.



          The answer is "because $f$ is not defined on all of $Bbb R^2$ --- for certain $(x,y)$ pairs, $f(x, y)$ is undefined, and those pairs cannot be part of the solution to equation (1)."



          For instance, let's look at the point $(a, b) = (-1, -1)$. Clearly this point satisfies $g(a, b) = a - b = (-1) - (-1) = 0$. But what is $f(a, b)$? Well, your formula for $f$ is
          $$
          f(x, y) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)
          $$

          and plugging in $-1$ for each of $x$ and $y$ gives, deep in the middle, this:
          $$
          ldots left(fracleft((-1)sqrtfracleft-(-1)-5+5right)5right)^2 ldots
          $$

          and the middle square root in that simplifies to
          beginalign
          &(-1)sqrtfracleft-(-1)-5\
          &= -sqrtfracleft1-5\
          &= -sqrtfrac-4\
          &= -sqrtfrac4-4\
          &= -sqrt-1
          endalign

          which is undefined. So because $f(a, b)$ is undefined, so is $f(a, b) cdot g(a, b)$, and hence this product cannot be zero...so the point $(a, b) = (-1, -1)$ is not part of the plot of the solutions to equation (1).



          The modified theorem that gets to what's going on here is this:




          Theorem: Suppose $D, E subset Bbb R^2$, $f:D to Bbb R$, $g:E to Bbb R$, $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
          set of the equation $f(x, y) = 0$. Then the solution set of the
          equation $$ f(x,y) g(x, y) = 0 $$ is exactly $(Fcap E) cup (G cap D)$.




          which says that only those solutions of $g = 0$ (i.e., $G$) that lie in the domain of $f$ (i.e., $D$) get included in the result, and only solutions of $f = 0$ that lie in the domain of $E$ get included in the result.






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            Thank you! I understand. But don't they do the same thing in viral batman logo equation? With all the complex number mess and things ? How is it fine there then?
            $endgroup$
            – user68153
            Mar 26 at 15:23










          • $begingroup$
            You might want to look at copper.hat's answer to this question (math.stackexchange.com/questions/54506/…) about the Batman equation. As an alternative, you can consider that your equation (and/or the batman equation) is defined by a product of functions $f,g: Bbb C^2 to Bbb R$ (with some slight problems where denominators are zero), and the graph you're hoping to get is the intersection of the solutions of $fg = 0$ (in $Bbb C^2$) with the real plane $Bbb R^2 subset Bbb C^2$. It's hardly surprising that Desmos, etc., cannot guess this. :)
            $endgroup$
            – John Hughes
            Mar 26 at 17:42










          • $begingroup$
            So even the batman equation doesn't work in decent plotters? Whoa! All the excitement!
            $endgroup$
            – user68153
            Mar 27 at 2:32


















          0












          $begingroup$

          Notice that if $f(x) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)$



          then $f(x)$ is not defined for $x=-5$ because then $sqrt frac -5-x-5$ is dividing by $0$. And if $x > -5$ then $-x-5 < 0$ so $sqrt frac -5-x-5$ is the square root of a negative number.



          So $f(x)$ simply doesn't work and does not exist if $x ge -5$.



          And if $f(x)$ does not exist for $x ge -5$ then $f(x)cdot (x-y)$ can't exist for $x ge -5$ either.



          It doesn't matter that multiplying by $f(x)(x-y) = 0$ and $0$ exists always. If $f(x)$ doesn't exist then neither does $f(x)times something else$. So we have $f(x)(x-y)= 0$ AND $x < -5$. We can't magic things into existence just because they "cancel out".






          share|cite|improve this answer









          $endgroup$













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            2 Answers
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            active

            oldest

            votes








            2 Answers
            2






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes









            1












            $begingroup$

            You're completely right about "the other equations" being the source of the problem.



            Let me start with something simple. What is the value of
            $$
            x cdot 0 ?
            $$



            It's very tempting to say, "It's $0$, of course, because anything times zero is zero!" But what if $x$ is, for instance, a cheese omelet? Then...well...then multiplication by zero isn't even defined, and you'd have to say "$x cdot 0$ is undefined" rather than "it's zero." You might say that "cheese omelet" isn't really math, but what if we replace $x$ like this:
            $$
            (1/0) cdot 0
            $$

            Then our first factor is undefined, and multiplication by undefined things is undefined, so once again the product is "undefined" rather than zero.



            Now let's look at a more complicated example. I'm going to stick with the real numbers -- no complex numbers allowed! -- and I want to look at
            $$
            sqrtx cdot 0
            $$

            You might say "Well, that's definitely zero!", but what if $x = -3$? Then the square root is undefined, and so the product is undefined as well.




            Here's a theorem that your question hints at:




            Theorem: Suppose $f, g : Bbb R^2 to Bbb R$, and $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $F cup G$.




            In your case, you've got $g(x, y) = x - y$, and the solutions for $x-y = 0$ form a line in the plane; that's the set $G$ in the theorem. You've multiplied it by some complicated mess that I'll temporarily call $f(x, y)$, and looked for solutions to
            $$
            tag1
            f(x, y) cdot g(x, y) = 0
            $$

            (perhaps using Desmos or some other graphing tool), and you're wondering why the solution set of this doesn't include all of $G$.



            The answer is "because $f$ is not defined on all of $Bbb R^2$ --- for certain $(x,y)$ pairs, $f(x, y)$ is undefined, and those pairs cannot be part of the solution to equation (1)."



            For instance, let's look at the point $(a, b) = (-1, -1)$. Clearly this point satisfies $g(a, b) = a - b = (-1) - (-1) = 0$. But what is $f(a, b)$? Well, your formula for $f$ is
            $$
            f(x, y) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)
            $$

            and plugging in $-1$ for each of $x$ and $y$ gives, deep in the middle, this:
            $$
            ldots left(fracleft((-1)sqrtfracleft-(-1)-5+5right)5right)^2 ldots
            $$

            and the middle square root in that simplifies to
            beginalign
            &(-1)sqrtfracleft-(-1)-5\
            &= -sqrtfracleft1-5\
            &= -sqrtfrac-4\
            &= -sqrtfrac4-4\
            &= -sqrt-1
            endalign

            which is undefined. So because $f(a, b)$ is undefined, so is $f(a, b) cdot g(a, b)$, and hence this product cannot be zero...so the point $(a, b) = (-1, -1)$ is not part of the plot of the solutions to equation (1).



            The modified theorem that gets to what's going on here is this:




            Theorem: Suppose $D, E subset Bbb R^2$, $f:D to Bbb R$, $g:E to Bbb R$, $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $(Fcap E) cup (G cap D)$.




            which says that only those solutions of $g = 0$ (i.e., $G$) that lie in the domain of $f$ (i.e., $D$) get included in the result, and only solutions of $f = 0$ that lie in the domain of $E$ get included in the result.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              Thank you! I understand. But don't they do the same thing in viral batman logo equation? With all the complex number mess and things ? How is it fine there then?
              $endgroup$
              – user68153
              Mar 26 at 15:23










            • $begingroup$
              You might want to look at copper.hat's answer to this question (math.stackexchange.com/questions/54506/…) about the Batman equation. As an alternative, you can consider that your equation (and/or the batman equation) is defined by a product of functions $f,g: Bbb C^2 to Bbb R$ (with some slight problems where denominators are zero), and the graph you're hoping to get is the intersection of the solutions of $fg = 0$ (in $Bbb C^2$) with the real plane $Bbb R^2 subset Bbb C^2$. It's hardly surprising that Desmos, etc., cannot guess this. :)
              $endgroup$
              – John Hughes
              Mar 26 at 17:42










            • $begingroup$
              So even the batman equation doesn't work in decent plotters? Whoa! All the excitement!
              $endgroup$
              – user68153
              Mar 27 at 2:32















            1












            $begingroup$

            You're completely right about "the other equations" being the source of the problem.



            Let me start with something simple. What is the value of
            $$
            x cdot 0 ?
            $$



            It's very tempting to say, "It's $0$, of course, because anything times zero is zero!" But what if $x$ is, for instance, a cheese omelet? Then...well...then multiplication by zero isn't even defined, and you'd have to say "$x cdot 0$ is undefined" rather than "it's zero." You might say that "cheese omelet" isn't really math, but what if we replace $x$ like this:
            $$
            (1/0) cdot 0
            $$

            Then our first factor is undefined, and multiplication by undefined things is undefined, so once again the product is "undefined" rather than zero.



            Now let's look at a more complicated example. I'm going to stick with the real numbers -- no complex numbers allowed! -- and I want to look at
            $$
            sqrtx cdot 0
            $$

            You might say "Well, that's definitely zero!", but what if $x = -3$? Then the square root is undefined, and so the product is undefined as well.




            Here's a theorem that your question hints at:




            Theorem: Suppose $f, g : Bbb R^2 to Bbb R$, and $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $F cup G$.




            In your case, you've got $g(x, y) = x - y$, and the solutions for $x-y = 0$ form a line in the plane; that's the set $G$ in the theorem. You've multiplied it by some complicated mess that I'll temporarily call $f(x, y)$, and looked for solutions to
            $$
            tag1
            f(x, y) cdot g(x, y) = 0
            $$

            (perhaps using Desmos or some other graphing tool), and you're wondering why the solution set of this doesn't include all of $G$.



            The answer is "because $f$ is not defined on all of $Bbb R^2$ --- for certain $(x,y)$ pairs, $f(x, y)$ is undefined, and those pairs cannot be part of the solution to equation (1)."



            For instance, let's look at the point $(a, b) = (-1, -1)$. Clearly this point satisfies $g(a, b) = a - b = (-1) - (-1) = 0$. But what is $f(a, b)$? Well, your formula for $f$ is
            $$
            f(x, y) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)
            $$

            and plugging in $-1$ for each of $x$ and $y$ gives, deep in the middle, this:
            $$
            ldots left(fracleft((-1)sqrtfracleft-(-1)-5+5right)5right)^2 ldots
            $$

            and the middle square root in that simplifies to
            beginalign
            &(-1)sqrtfracleft-(-1)-5\
            &= -sqrtfracleft1-5\
            &= -sqrtfrac-4\
            &= -sqrtfrac4-4\
            &= -sqrt-1
            endalign

            which is undefined. So because $f(a, b)$ is undefined, so is $f(a, b) cdot g(a, b)$, and hence this product cannot be zero...so the point $(a, b) = (-1, -1)$ is not part of the plot of the solutions to equation (1).



            The modified theorem that gets to what's going on here is this:




            Theorem: Suppose $D, E subset Bbb R^2$, $f:D to Bbb R$, $g:E to Bbb R$, $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $(Fcap E) cup (G cap D)$.




            which says that only those solutions of $g = 0$ (i.e., $G$) that lie in the domain of $f$ (i.e., $D$) get included in the result, and only solutions of $f = 0$ that lie in the domain of $E$ get included in the result.






            share|cite|improve this answer











            $endgroup$












            • $begingroup$
              Thank you! I understand. But don't they do the same thing in viral batman logo equation? With all the complex number mess and things ? How is it fine there then?
              $endgroup$
              – user68153
              Mar 26 at 15:23










            • $begingroup$
              You might want to look at copper.hat's answer to this question (math.stackexchange.com/questions/54506/…) about the Batman equation. As an alternative, you can consider that your equation (and/or the batman equation) is defined by a product of functions $f,g: Bbb C^2 to Bbb R$ (with some slight problems where denominators are zero), and the graph you're hoping to get is the intersection of the solutions of $fg = 0$ (in $Bbb C^2$) with the real plane $Bbb R^2 subset Bbb C^2$. It's hardly surprising that Desmos, etc., cannot guess this. :)
              $endgroup$
              – John Hughes
              Mar 26 at 17:42










            • $begingroup$
              So even the batman equation doesn't work in decent plotters? Whoa! All the excitement!
              $endgroup$
              – user68153
              Mar 27 at 2:32













            1












            1








            1





            $begingroup$

            You're completely right about "the other equations" being the source of the problem.



            Let me start with something simple. What is the value of
            $$
            x cdot 0 ?
            $$



            It's very tempting to say, "It's $0$, of course, because anything times zero is zero!" But what if $x$ is, for instance, a cheese omelet? Then...well...then multiplication by zero isn't even defined, and you'd have to say "$x cdot 0$ is undefined" rather than "it's zero." You might say that "cheese omelet" isn't really math, but what if we replace $x$ like this:
            $$
            (1/0) cdot 0
            $$

            Then our first factor is undefined, and multiplication by undefined things is undefined, so once again the product is "undefined" rather than zero.



            Now let's look at a more complicated example. I'm going to stick with the real numbers -- no complex numbers allowed! -- and I want to look at
            $$
            sqrtx cdot 0
            $$

            You might say "Well, that's definitely zero!", but what if $x = -3$? Then the square root is undefined, and so the product is undefined as well.




            Here's a theorem that your question hints at:




            Theorem: Suppose $f, g : Bbb R^2 to Bbb R$, and $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $F cup G$.




            In your case, you've got $g(x, y) = x - y$, and the solutions for $x-y = 0$ form a line in the plane; that's the set $G$ in the theorem. You've multiplied it by some complicated mess that I'll temporarily call $f(x, y)$, and looked for solutions to
            $$
            tag1
            f(x, y) cdot g(x, y) = 0
            $$

            (perhaps using Desmos or some other graphing tool), and you're wondering why the solution set of this doesn't include all of $G$.



            The answer is "because $f$ is not defined on all of $Bbb R^2$ --- for certain $(x,y)$ pairs, $f(x, y)$ is undefined, and those pairs cannot be part of the solution to equation (1)."



            For instance, let's look at the point $(a, b) = (-1, -1)$. Clearly this point satisfies $g(a, b) = a - b = (-1) - (-1) = 0$. But what is $f(a, b)$? Well, your formula for $f$ is
            $$
            f(x, y) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)
            $$

            and plugging in $-1$ for each of $x$ and $y$ gives, deep in the middle, this:
            $$
            ldots left(fracleft((-1)sqrtfracleft-(-1)-5+5right)5right)^2 ldots
            $$

            and the middle square root in that simplifies to
            beginalign
            &(-1)sqrtfracleft-(-1)-5\
            &= -sqrtfracleft1-5\
            &= -sqrtfrac-4\
            &= -sqrtfrac4-4\
            &= -sqrt-1
            endalign

            which is undefined. So because $f(a, b)$ is undefined, so is $f(a, b) cdot g(a, b)$, and hence this product cannot be zero...so the point $(a, b) = (-1, -1)$ is not part of the plot of the solutions to equation (1).



            The modified theorem that gets to what's going on here is this:




            Theorem: Suppose $D, E subset Bbb R^2$, $f:D to Bbb R$, $g:E to Bbb R$, $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $(Fcap E) cup (G cap D)$.




            which says that only those solutions of $g = 0$ (i.e., $G$) that lie in the domain of $f$ (i.e., $D$) get included in the result, and only solutions of $f = 0$ that lie in the domain of $E$ get included in the result.






            share|cite|improve this answer











            $endgroup$



            You're completely right about "the other equations" being the source of the problem.



            Let me start with something simple. What is the value of
            $$
            x cdot 0 ?
            $$



            It's very tempting to say, "It's $0$, of course, because anything times zero is zero!" But what if $x$ is, for instance, a cheese omelet? Then...well...then multiplication by zero isn't even defined, and you'd have to say "$x cdot 0$ is undefined" rather than "it's zero." You might say that "cheese omelet" isn't really math, but what if we replace $x$ like this:
            $$
            (1/0) cdot 0
            $$

            Then our first factor is undefined, and multiplication by undefined things is undefined, so once again the product is "undefined" rather than zero.



            Now let's look at a more complicated example. I'm going to stick with the real numbers -- no complex numbers allowed! -- and I want to look at
            $$
            sqrtx cdot 0
            $$

            You might say "Well, that's definitely zero!", but what if $x = -3$? Then the square root is undefined, and so the product is undefined as well.




            Here's a theorem that your question hints at:




            Theorem: Suppose $f, g : Bbb R^2 to Bbb R$, and $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $F cup G$.




            In your case, you've got $g(x, y) = x - y$, and the solutions for $x-y = 0$ form a line in the plane; that's the set $G$ in the theorem. You've multiplied it by some complicated mess that I'll temporarily call $f(x, y)$, and looked for solutions to
            $$
            tag1
            f(x, y) cdot g(x, y) = 0
            $$

            (perhaps using Desmos or some other graphing tool), and you're wondering why the solution set of this doesn't include all of $G$.



            The answer is "because $f$ is not defined on all of $Bbb R^2$ --- for certain $(x,y)$ pairs, $f(x, y)$ is undefined, and those pairs cannot be part of the solution to equation (1)."



            For instance, let's look at the point $(a, b) = (-1, -1)$. Clearly this point satisfies $g(a, b) = a - b = (-1) - (-1) = 0$. But what is $f(a, b)$? Well, your formula for $f$ is
            $$
            f(x, y) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)
            $$

            and plugging in $-1$ for each of $x$ and $y$ gives, deep in the middle, this:
            $$
            ldots left(fracleft((-1)sqrtfracleft-(-1)-5+5right)5right)^2 ldots
            $$

            and the middle square root in that simplifies to
            beginalign
            &(-1)sqrtfracleft-(-1)-5\
            &= -sqrtfracleft1-5\
            &= -sqrtfrac-4\
            &= -sqrtfrac4-4\
            &= -sqrt-1
            endalign

            which is undefined. So because $f(a, b)$ is undefined, so is $f(a, b) cdot g(a, b)$, and hence this product cannot be zero...so the point $(a, b) = (-1, -1)$ is not part of the plot of the solutions to equation (1).



            The modified theorem that gets to what's going on here is this:




            Theorem: Suppose $D, E subset Bbb R^2$, $f:D to Bbb R$, $g:E to Bbb R$, $G$ is the solution set of the equation $g(x, y) = 0$, and $F$ is the solution
            set of the equation $f(x, y) = 0$. Then the solution set of the
            equation $$ f(x,y) g(x, y) = 0 $$ is exactly $(Fcap E) cup (G cap D)$.




            which says that only those solutions of $g = 0$ (i.e., $G$) that lie in the domain of $f$ (i.e., $D$) get included in the result, and only solutions of $f = 0$ that lie in the domain of $E$ get included in the result.







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Mar 26 at 22:16

























            answered Mar 26 at 14:43









            John HughesJohn Hughes

            65.5k24293




            65.5k24293











            • $begingroup$
              Thank you! I understand. But don't they do the same thing in viral batman logo equation? With all the complex number mess and things ? How is it fine there then?
              $endgroup$
              – user68153
              Mar 26 at 15:23










            • $begingroup$
              You might want to look at copper.hat's answer to this question (math.stackexchange.com/questions/54506/…) about the Batman equation. As an alternative, you can consider that your equation (and/or the batman equation) is defined by a product of functions $f,g: Bbb C^2 to Bbb R$ (with some slight problems where denominators are zero), and the graph you're hoping to get is the intersection of the solutions of $fg = 0$ (in $Bbb C^2$) with the real plane $Bbb R^2 subset Bbb C^2$. It's hardly surprising that Desmos, etc., cannot guess this. :)
              $endgroup$
              – John Hughes
              Mar 26 at 17:42










            • $begingroup$
              So even the batman equation doesn't work in decent plotters? Whoa! All the excitement!
              $endgroup$
              – user68153
              Mar 27 at 2:32
















            • $begingroup$
              Thank you! I understand. But don't they do the same thing in viral batman logo equation? With all the complex number mess and things ? How is it fine there then?
              $endgroup$
              – user68153
              Mar 26 at 15:23










            • $begingroup$
              You might want to look at copper.hat's answer to this question (math.stackexchange.com/questions/54506/…) about the Batman equation. As an alternative, you can consider that your equation (and/or the batman equation) is defined by a product of functions $f,g: Bbb C^2 to Bbb R$ (with some slight problems where denominators are zero), and the graph you're hoping to get is the intersection of the solutions of $fg = 0$ (in $Bbb C^2$) with the real plane $Bbb R^2 subset Bbb C^2$. It's hardly surprising that Desmos, etc., cannot guess this. :)
              $endgroup$
              – John Hughes
              Mar 26 at 17:42










            • $begingroup$
              So even the batman equation doesn't work in decent plotters? Whoa! All the excitement!
              $endgroup$
              – user68153
              Mar 27 at 2:32















            $begingroup$
            Thank you! I understand. But don't they do the same thing in viral batman logo equation? With all the complex number mess and things ? How is it fine there then?
            $endgroup$
            – user68153
            Mar 26 at 15:23




            $begingroup$
            Thank you! I understand. But don't they do the same thing in viral batman logo equation? With all the complex number mess and things ? How is it fine there then?
            $endgroup$
            – user68153
            Mar 26 at 15:23












            $begingroup$
            You might want to look at copper.hat's answer to this question (math.stackexchange.com/questions/54506/…) about the Batman equation. As an alternative, you can consider that your equation (and/or the batman equation) is defined by a product of functions $f,g: Bbb C^2 to Bbb R$ (with some slight problems where denominators are zero), and the graph you're hoping to get is the intersection of the solutions of $fg = 0$ (in $Bbb C^2$) with the real plane $Bbb R^2 subset Bbb C^2$. It's hardly surprising that Desmos, etc., cannot guess this. :)
            $endgroup$
            – John Hughes
            Mar 26 at 17:42




            $begingroup$
            You might want to look at copper.hat's answer to this question (math.stackexchange.com/questions/54506/…) about the Batman equation. As an alternative, you can consider that your equation (and/or the batman equation) is defined by a product of functions $f,g: Bbb C^2 to Bbb R$ (with some slight problems where denominators are zero), and the graph you're hoping to get is the intersection of the solutions of $fg = 0$ (in $Bbb C^2$) with the real plane $Bbb R^2 subset Bbb C^2$. It's hardly surprising that Desmos, etc., cannot guess this. :)
            $endgroup$
            – John Hughes
            Mar 26 at 17:42












            $begingroup$
            So even the batman equation doesn't work in decent plotters? Whoa! All the excitement!
            $endgroup$
            – user68153
            Mar 27 at 2:32




            $begingroup$
            So even the batman equation doesn't work in decent plotters? Whoa! All the excitement!
            $endgroup$
            – user68153
            Mar 27 at 2:32











            0












            $begingroup$

            Notice that if $f(x) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)$



            then $f(x)$ is not defined for $x=-5$ because then $sqrt frac -5-x-5$ is dividing by $0$. And if $x > -5$ then $-x-5 < 0$ so $sqrt frac -5-x-5$ is the square root of a negative number.



            So $f(x)$ simply doesn't work and does not exist if $x ge -5$.



            And if $f(x)$ does not exist for $x ge -5$ then $f(x)cdot (x-y)$ can't exist for $x ge -5$ either.



            It doesn't matter that multiplying by $f(x)(x-y) = 0$ and $0$ exists always. If $f(x)$ doesn't exist then neither does $f(x)times something else$. So we have $f(x)(x-y)= 0$ AND $x < -5$. We can't magic things into existence just because they "cancel out".






            share|cite|improve this answer









            $endgroup$

















              0












              $begingroup$

              Notice that if $f(x) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)$



              then $f(x)$ is not defined for $x=-5$ because then $sqrt frac -5-x-5$ is dividing by $0$. And if $x > -5$ then $-x-5 < 0$ so $sqrt frac -5-x-5$ is the square root of a negative number.



              So $f(x)$ simply doesn't work and does not exist if $x ge -5$.



              And if $f(x)$ does not exist for $x ge -5$ then $f(x)cdot (x-y)$ can't exist for $x ge -5$ either.



              It doesn't matter that multiplying by $f(x)(x-y) = 0$ and $0$ exists always. If $f(x)$ doesn't exist then neither does $f(x)times something else$. So we have $f(x)(x-y)= 0$ AND $x < -5$. We can't magic things into existence just because they "cancel out".






              share|cite|improve this answer









              $endgroup$















                0












                0








                0





                $begingroup$

                Notice that if $f(x) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)$



                then $f(x)$ is not defined for $x=-5$ because then $sqrt frac -5-x-5$ is dividing by $0$. And if $x > -5$ then $-x-5 < 0$ so $sqrt frac -5-x-5$ is the square root of a negative number.



                So $f(x)$ simply doesn't work and does not exist if $x ge -5$.



                And if $f(x)$ does not exist for $x ge -5$ then $f(x)cdot (x-y)$ can't exist for $x ge -5$ either.



                It doesn't matter that multiplying by $f(x)(x-y) = 0$ and $0$ exists always. If $f(x)$ doesn't exist then neither does $f(x)times something else$. So we have $f(x)(x-y)= 0$ AND $x < -5$. We can't magic things into existence just because they "cancel out".






                share|cite|improve this answer









                $endgroup$



                Notice that if $f(x) = left(left(fracleft(xsqrtfracleft-x-5+5right)5right)^2+left(fracy4right)^2-1right)left(left(fracleft(xsqrtfracleft-x-5+5right)3.4right)^2+left(fracy2.72right)^2-1right)$



                then $f(x)$ is not defined for $x=-5$ because then $sqrt frac -5-x-5$ is dividing by $0$. And if $x > -5$ then $-x-5 < 0$ so $sqrt frac -5-x-5$ is the square root of a negative number.



                So $f(x)$ simply doesn't work and does not exist if $x ge -5$.



                And if $f(x)$ does not exist for $x ge -5$ then $f(x)cdot (x-y)$ can't exist for $x ge -5$ either.



                It doesn't matter that multiplying by $f(x)(x-y) = 0$ and $0$ exists always. If $f(x)$ doesn't exist then neither does $f(x)times something else$. So we have $f(x)(x-y)= 0$ AND $x < -5$. We can't magic things into existence just because they "cancel out".







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Mar 26 at 22:41









                fleabloodfleablood

                1




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