How do you solve these 2 equations?











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$$xy = 1/6$$
$$y+x = 5xy$$
I tried solving them using all methods - substitution, elimination and graphing - but can't get the solutions










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  • What were "all those methods" that you tried?
    – Parcly Taxel
    Nov 18 at 6:07










  • Substitution, elimination and graphing.
    – Jullian Santos
    Nov 18 at 6:11










  • Take the first equation and insert it into the second
    – Fakemistake
    Nov 18 at 6:11










  • Actually I can't use the graphing so what I meant was substitution and elimination.
    – Jullian Santos
    Nov 18 at 6:12










  • Remember if the amount of variables and amount of equations are equal, you can always solve that system of equations. Just substitute the value of any one variable from one equation in another.
    – PradyumanDixit
    Nov 18 at 6:12















up vote
0
down vote

favorite












$$xy = 1/6$$
$$y+x = 5xy$$
I tried solving them using all methods - substitution, elimination and graphing - but can't get the solutions










share|cite|improve this question
























  • What were "all those methods" that you tried?
    – Parcly Taxel
    Nov 18 at 6:07










  • Substitution, elimination and graphing.
    – Jullian Santos
    Nov 18 at 6:11










  • Take the first equation and insert it into the second
    – Fakemistake
    Nov 18 at 6:11










  • Actually I can't use the graphing so what I meant was substitution and elimination.
    – Jullian Santos
    Nov 18 at 6:12










  • Remember if the amount of variables and amount of equations are equal, you can always solve that system of equations. Just substitute the value of any one variable from one equation in another.
    – PradyumanDixit
    Nov 18 at 6:12













up vote
0
down vote

favorite









up vote
0
down vote

favorite











$$xy = 1/6$$
$$y+x = 5xy$$
I tried solving them using all methods - substitution, elimination and graphing - but can't get the solutions










share|cite|improve this question















$$xy = 1/6$$
$$y+x = 5xy$$
I tried solving them using all methods - substitution, elimination and graphing - but can't get the solutions







systems-of-equations quadratics symmetric-polynomials






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share|cite|improve this question













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edited Nov 23 at 11:59









Harry Peter

5,43111439




5,43111439










asked Nov 18 at 6:06









Jullian Santos

6




6












  • What were "all those methods" that you tried?
    – Parcly Taxel
    Nov 18 at 6:07










  • Substitution, elimination and graphing.
    – Jullian Santos
    Nov 18 at 6:11










  • Take the first equation and insert it into the second
    – Fakemistake
    Nov 18 at 6:11










  • Actually I can't use the graphing so what I meant was substitution and elimination.
    – Jullian Santos
    Nov 18 at 6:12










  • Remember if the amount of variables and amount of equations are equal, you can always solve that system of equations. Just substitute the value of any one variable from one equation in another.
    – PradyumanDixit
    Nov 18 at 6:12


















  • What were "all those methods" that you tried?
    – Parcly Taxel
    Nov 18 at 6:07










  • Substitution, elimination and graphing.
    – Jullian Santos
    Nov 18 at 6:11










  • Take the first equation and insert it into the second
    – Fakemistake
    Nov 18 at 6:11










  • Actually I can't use the graphing so what I meant was substitution and elimination.
    – Jullian Santos
    Nov 18 at 6:12










  • Remember if the amount of variables and amount of equations are equal, you can always solve that system of equations. Just substitute the value of any one variable from one equation in another.
    – PradyumanDixit
    Nov 18 at 6:12
















What were "all those methods" that you tried?
– Parcly Taxel
Nov 18 at 6:07




What were "all those methods" that you tried?
– Parcly Taxel
Nov 18 at 6:07












Substitution, elimination and graphing.
– Jullian Santos
Nov 18 at 6:11




Substitution, elimination and graphing.
– Jullian Santos
Nov 18 at 6:11












Take the first equation and insert it into the second
– Fakemistake
Nov 18 at 6:11




Take the first equation and insert it into the second
– Fakemistake
Nov 18 at 6:11












Actually I can't use the graphing so what I meant was substitution and elimination.
– Jullian Santos
Nov 18 at 6:12




Actually I can't use the graphing so what I meant was substitution and elimination.
– Jullian Santos
Nov 18 at 6:12












Remember if the amount of variables and amount of equations are equal, you can always solve that system of equations. Just substitute the value of any one variable from one equation in another.
– PradyumanDixit
Nov 18 at 6:12




Remember if the amount of variables and amount of equations are equal, you can always solve that system of equations. Just substitute the value of any one variable from one equation in another.
– PradyumanDixit
Nov 18 at 6:12










2 Answers
2






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oldest

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up vote
2
down vote













You can solve this equation by using substitution of variables.



Using the first equation $xy=1/6$, we can rewrite it as $y=frac1{6x}$.



Plugging this in to the second equation, we get $$x+frac1{6x}=5frac{x}{6x}$$



Simplify the right-hand side to $frac56$ and multiply both sides by $x$ to obtain



$$x^2+frac16=frac56x to x^2-frac56x+frac16=0$$



Using the quadratic formula, we now have $$x=frac{frac56 pm sqrt{frac{25}{36}-frac46}}{2}=frac5{12} pm frac12sqrt{frac1{36}}=frac5{12} pm frac1{12}$$



Our solutions for $x$ are $frac13$ and $frac12$. Using $y=frac1{6x}$, we get the coordinate pairs to be $(frac13,frac12)$ and $(frac12,frac13)$.






share|cite|improve this answer

















  • 1




    In the quadratic equation, it's easier to compute if we just first multiply both sides with $6$ to get $6x^2 -5x +1=0$, and get only integers to compute with. The determinant becomes $(-5)^2 - 4cdot 6 = 1$ so we get a nice result.
    – Henno Brandsma
    Nov 18 at 6:27










  • Thanks for the answer!
    – Jullian Santos
    Nov 18 at 6:27


















up vote
0
down vote













1) Substitute xy=$frac{1}{6}$ into the 2nd equation to get x+y=$frac{5}{6}$



2) Solving xy=$frac{1}{6}$ and x+y=$frac{5}{6}$ is equivalent to finding the zeros of the function f(z)=$z^2$+$frac{5}{6}$z+$frac{1}{6}$=0, so using the quadratic formula, x=$-frac{1}{3}$ & y=$-frac{1}{2}$ or y=$-frac{1}{3}$ & x=$-frac{1}{2}$.






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    2 Answers
    2






    active

    oldest

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    2 Answers
    2






    active

    oldest

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    active

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    active

    oldest

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    up vote
    2
    down vote













    You can solve this equation by using substitution of variables.



    Using the first equation $xy=1/6$, we can rewrite it as $y=frac1{6x}$.



    Plugging this in to the second equation, we get $$x+frac1{6x}=5frac{x}{6x}$$



    Simplify the right-hand side to $frac56$ and multiply both sides by $x$ to obtain



    $$x^2+frac16=frac56x to x^2-frac56x+frac16=0$$



    Using the quadratic formula, we now have $$x=frac{frac56 pm sqrt{frac{25}{36}-frac46}}{2}=frac5{12} pm frac12sqrt{frac1{36}}=frac5{12} pm frac1{12}$$



    Our solutions for $x$ are $frac13$ and $frac12$. Using $y=frac1{6x}$, we get the coordinate pairs to be $(frac13,frac12)$ and $(frac12,frac13)$.






    share|cite|improve this answer

















    • 1




      In the quadratic equation, it's easier to compute if we just first multiply both sides with $6$ to get $6x^2 -5x +1=0$, and get only integers to compute with. The determinant becomes $(-5)^2 - 4cdot 6 = 1$ so we get a nice result.
      – Henno Brandsma
      Nov 18 at 6:27










    • Thanks for the answer!
      – Jullian Santos
      Nov 18 at 6:27















    up vote
    2
    down vote













    You can solve this equation by using substitution of variables.



    Using the first equation $xy=1/6$, we can rewrite it as $y=frac1{6x}$.



    Plugging this in to the second equation, we get $$x+frac1{6x}=5frac{x}{6x}$$



    Simplify the right-hand side to $frac56$ and multiply both sides by $x$ to obtain



    $$x^2+frac16=frac56x to x^2-frac56x+frac16=0$$



    Using the quadratic formula, we now have $$x=frac{frac56 pm sqrt{frac{25}{36}-frac46}}{2}=frac5{12} pm frac12sqrt{frac1{36}}=frac5{12} pm frac1{12}$$



    Our solutions for $x$ are $frac13$ and $frac12$. Using $y=frac1{6x}$, we get the coordinate pairs to be $(frac13,frac12)$ and $(frac12,frac13)$.






    share|cite|improve this answer

















    • 1




      In the quadratic equation, it's easier to compute if we just first multiply both sides with $6$ to get $6x^2 -5x +1=0$, and get only integers to compute with. The determinant becomes $(-5)^2 - 4cdot 6 = 1$ so we get a nice result.
      – Henno Brandsma
      Nov 18 at 6:27










    • Thanks for the answer!
      – Jullian Santos
      Nov 18 at 6:27













    up vote
    2
    down vote










    up vote
    2
    down vote









    You can solve this equation by using substitution of variables.



    Using the first equation $xy=1/6$, we can rewrite it as $y=frac1{6x}$.



    Plugging this in to the second equation, we get $$x+frac1{6x}=5frac{x}{6x}$$



    Simplify the right-hand side to $frac56$ and multiply both sides by $x$ to obtain



    $$x^2+frac16=frac56x to x^2-frac56x+frac16=0$$



    Using the quadratic formula, we now have $$x=frac{frac56 pm sqrt{frac{25}{36}-frac46}}{2}=frac5{12} pm frac12sqrt{frac1{36}}=frac5{12} pm frac1{12}$$



    Our solutions for $x$ are $frac13$ and $frac12$. Using $y=frac1{6x}$, we get the coordinate pairs to be $(frac13,frac12)$ and $(frac12,frac13)$.






    share|cite|improve this answer












    You can solve this equation by using substitution of variables.



    Using the first equation $xy=1/6$, we can rewrite it as $y=frac1{6x}$.



    Plugging this in to the second equation, we get $$x+frac1{6x}=5frac{x}{6x}$$



    Simplify the right-hand side to $frac56$ and multiply both sides by $x$ to obtain



    $$x^2+frac16=frac56x to x^2-frac56x+frac16=0$$



    Using the quadratic formula, we now have $$x=frac{frac56 pm sqrt{frac{25}{36}-frac46}}{2}=frac5{12} pm frac12sqrt{frac1{36}}=frac5{12} pm frac1{12}$$



    Our solutions for $x$ are $frac13$ and $frac12$. Using $y=frac1{6x}$, we get the coordinate pairs to be $(frac13,frac12)$ and $(frac12,frac13)$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Nov 18 at 6:20









    Christopher Marley

    86815




    86815








    • 1




      In the quadratic equation, it's easier to compute if we just first multiply both sides with $6$ to get $6x^2 -5x +1=0$, and get only integers to compute with. The determinant becomes $(-5)^2 - 4cdot 6 = 1$ so we get a nice result.
      – Henno Brandsma
      Nov 18 at 6:27










    • Thanks for the answer!
      – Jullian Santos
      Nov 18 at 6:27














    • 1




      In the quadratic equation, it's easier to compute if we just first multiply both sides with $6$ to get $6x^2 -5x +1=0$, and get only integers to compute with. The determinant becomes $(-5)^2 - 4cdot 6 = 1$ so we get a nice result.
      – Henno Brandsma
      Nov 18 at 6:27










    • Thanks for the answer!
      – Jullian Santos
      Nov 18 at 6:27








    1




    1




    In the quadratic equation, it's easier to compute if we just first multiply both sides with $6$ to get $6x^2 -5x +1=0$, and get only integers to compute with. The determinant becomes $(-5)^2 - 4cdot 6 = 1$ so we get a nice result.
    – Henno Brandsma
    Nov 18 at 6:27




    In the quadratic equation, it's easier to compute if we just first multiply both sides with $6$ to get $6x^2 -5x +1=0$, and get only integers to compute with. The determinant becomes $(-5)^2 - 4cdot 6 = 1$ so we get a nice result.
    – Henno Brandsma
    Nov 18 at 6:27












    Thanks for the answer!
    – Jullian Santos
    Nov 18 at 6:27




    Thanks for the answer!
    – Jullian Santos
    Nov 18 at 6:27










    up vote
    0
    down vote













    1) Substitute xy=$frac{1}{6}$ into the 2nd equation to get x+y=$frac{5}{6}$



    2) Solving xy=$frac{1}{6}$ and x+y=$frac{5}{6}$ is equivalent to finding the zeros of the function f(z)=$z^2$+$frac{5}{6}$z+$frac{1}{6}$=0, so using the quadratic formula, x=$-frac{1}{3}$ & y=$-frac{1}{2}$ or y=$-frac{1}{3}$ & x=$-frac{1}{2}$.






    share|cite|improve this answer

























      up vote
      0
      down vote













      1) Substitute xy=$frac{1}{6}$ into the 2nd equation to get x+y=$frac{5}{6}$



      2) Solving xy=$frac{1}{6}$ and x+y=$frac{5}{6}$ is equivalent to finding the zeros of the function f(z)=$z^2$+$frac{5}{6}$z+$frac{1}{6}$=0, so using the quadratic formula, x=$-frac{1}{3}$ & y=$-frac{1}{2}$ or y=$-frac{1}{3}$ & x=$-frac{1}{2}$.






      share|cite|improve this answer























        up vote
        0
        down vote










        up vote
        0
        down vote









        1) Substitute xy=$frac{1}{6}$ into the 2nd equation to get x+y=$frac{5}{6}$



        2) Solving xy=$frac{1}{6}$ and x+y=$frac{5}{6}$ is equivalent to finding the zeros of the function f(z)=$z^2$+$frac{5}{6}$z+$frac{1}{6}$=0, so using the quadratic formula, x=$-frac{1}{3}$ & y=$-frac{1}{2}$ or y=$-frac{1}{3}$ & x=$-frac{1}{2}$.






        share|cite|improve this answer












        1) Substitute xy=$frac{1}{6}$ into the 2nd equation to get x+y=$frac{5}{6}$



        2) Solving xy=$frac{1}{6}$ and x+y=$frac{5}{6}$ is equivalent to finding the zeros of the function f(z)=$z^2$+$frac{5}{6}$z+$frac{1}{6}$=0, so using the quadratic formula, x=$-frac{1}{3}$ & y=$-frac{1}{2}$ or y=$-frac{1}{3}$ & x=$-frac{1}{2}$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Nov 18 at 6:23









        Muchang Bahng

        562




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