Solving an algebraic exercise using only one variable $x$












3












$begingroup$


This exercise is supposed to be solved using only one though I don't see how would that be possible.



Could someone tell how to solve using only one unknown variable $x$?



Thank you.



Exercise.
The sum of two numbers is $108$ and the double of the greater exceeds the triple of the minor in $156$. Find the numbers.



I solved using two variables $x$ and $y$ and specifying that $x>y$.






algebra-precalculus systems-of-equations







share|cite|improve this question









$endgroup$








  • 4




    $begingroup$
    Well, you can always replace $y$ with $108-x$ to eliminate a variable. Shouldn't change any of the algebra.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:53






  • 1




    $begingroup$
    You're right. how could I not realize about it. thank you @lulu
    $endgroup$
    – user459663
    Dec 15 '18 at 23:57






  • 1




    $begingroup$
    No problem. You were probably looking for some deep, meaningful trick. Alas, no.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:58






  • 1




    $begingroup$
    I dislike questions that specify the method to be used. When there are several you should be able to work with the one that suits your style best. If an instructor wants to test a particular method they should construct a problem for which that method is essentially the only way to go.
    $endgroup$
    – Ethan Bolker
    Dec 16 '18 at 1:37
















3












$begingroup$


This exercise is supposed to be solved using only one though I don't see how would that be possible.



Could someone tell how to solve using only one unknown variable $x$?



Thank you.



Exercise.
The sum of two numbers is $108$ and the double of the greater exceeds the triple of the minor in $156$. Find the numbers.



I solved using two variables $x$ and $y$ and specifying that $x>y$.






algebra-precalculus systems-of-equations







share|cite|improve this question









$endgroup$








  • 4




    $begingroup$
    Well, you can always replace $y$ with $108-x$ to eliminate a variable. Shouldn't change any of the algebra.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:53






  • 1




    $begingroup$
    You're right. how could I not realize about it. thank you @lulu
    $endgroup$
    – user459663
    Dec 15 '18 at 23:57






  • 1




    $begingroup$
    No problem. You were probably looking for some deep, meaningful trick. Alas, no.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:58






  • 1




    $begingroup$
    I dislike questions that specify the method to be used. When there are several you should be able to work with the one that suits your style best. If an instructor wants to test a particular method they should construct a problem for which that method is essentially the only way to go.
    $endgroup$
    – Ethan Bolker
    Dec 16 '18 at 1:37














3












3








3


1



$begingroup$


This exercise is supposed to be solved using only one though I don't see how would that be possible.



Could someone tell how to solve using only one unknown variable $x$?



Thank you.



Exercise.
The sum of two numbers is $108$ and the double of the greater exceeds the triple of the minor in $156$. Find the numbers.



I solved using two variables $x$ and $y$ and specifying that $x>y$.






algebra-precalculus systems-of-equations







share|cite|improve this question









$endgroup$




This exercise is supposed to be solved using only one though I don't see how would that be possible.



Could someone tell how to solve using only one unknown variable $x$?



Thank you.



Exercise.
The sum of two numbers is $108$ and the double of the greater exceeds the triple of the minor in $156$. Find the numbers.



I solved using two variables $x$ and $y$ and specifying that $x>y$.






algebra-precalculus systems-of-equations




algebra-precalculus systems-of-equations






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Dec 15 '18 at 23:50









user459663user459663

306




306








  • 4




    $begingroup$
    Well, you can always replace $y$ with $108-x$ to eliminate a variable. Shouldn't change any of the algebra.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:53






  • 1




    $begingroup$
    You're right. how could I not realize about it. thank you @lulu
    $endgroup$
    – user459663
    Dec 15 '18 at 23:57






  • 1




    $begingroup$
    No problem. You were probably looking for some deep, meaningful trick. Alas, no.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:58






  • 1




    $begingroup$
    I dislike questions that specify the method to be used. When there are several you should be able to work with the one that suits your style best. If an instructor wants to test a particular method they should construct a problem for which that method is essentially the only way to go.
    $endgroup$
    – Ethan Bolker
    Dec 16 '18 at 1:37














  • 4




    $begingroup$
    Well, you can always replace $y$ with $108-x$ to eliminate a variable. Shouldn't change any of the algebra.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:53






  • 1




    $begingroup$
    You're right. how could I not realize about it. thank you @lulu
    $endgroup$
    – user459663
    Dec 15 '18 at 23:57






  • 1




    $begingroup$
    No problem. You were probably looking for some deep, meaningful trick. Alas, no.
    $endgroup$
    – lulu
    Dec 15 '18 at 23:58






  • 1




    $begingroup$
    I dislike questions that specify the method to be used. When there are several you should be able to work with the one that suits your style best. If an instructor wants to test a particular method they should construct a problem for which that method is essentially the only way to go.
    $endgroup$
    – Ethan Bolker
    Dec 16 '18 at 1:37








4




4




$begingroup$
Well, you can always replace $y$ with $108-x$ to eliminate a variable. Shouldn't change any of the algebra.
$endgroup$
– lulu
Dec 15 '18 at 23:53




$begingroup$
Well, you can always replace $y$ with $108-x$ to eliminate a variable. Shouldn't change any of the algebra.
$endgroup$
– lulu
Dec 15 '18 at 23:53




1




1




$begingroup$
You're right. how could I not realize about it. thank you @lulu
$endgroup$
– user459663
Dec 15 '18 at 23:57




$begingroup$
You're right. how could I not realize about it. thank you @lulu
$endgroup$
– user459663
Dec 15 '18 at 23:57




1




1




$begingroup$
No problem. You were probably looking for some deep, meaningful trick. Alas, no.
$endgroup$
– lulu
Dec 15 '18 at 23:58




$begingroup$
No problem. You were probably looking for some deep, meaningful trick. Alas, no.
$endgroup$
– lulu
Dec 15 '18 at 23:58




1




1




$begingroup$
I dislike questions that specify the method to be used. When there are several you should be able to work with the one that suits your style best. If an instructor wants to test a particular method they should construct a problem for which that method is essentially the only way to go.
$endgroup$
– Ethan Bolker
Dec 16 '18 at 1:37




$begingroup$
I dislike questions that specify the method to be used. When there are several you should be able to work with the one that suits your style best. If an instructor wants to test a particular method they should construct a problem for which that method is essentially the only way to go.
$endgroup$
– Ethan Bolker
Dec 16 '18 at 1:37










2 Answers
2






active

oldest

votes


















3












$begingroup$

So let's piece this together:



"The sum of two numbers is $108$" - this gives us $x+y = 108$



"the double of the greater exceeds the triple of the minor in 156"- Without loss of generality, we can assume $x>y$, and thus see $2x - 3y = 156$.



Thus, we have a system of equations, which we seek to find $x$ and $y$:



$$x+y = 108$$
$$2x - 3y = 156$$



You could solve this by whatever method you prefer to do so. Since you seek to use only one variable, notice that the first equation implies $y = 108-x$, yielding



$$x+(108-x)= 108$$
$$2x - 3(108-x) = 156$$



This is a technique known as solving by substitution: solving for one of the variables and then substituting the resulting expression into the other equations.



Notice, too, if you used the second of the equations from our original system, you could have found



$$y = frac{156-2x}{-3}$$



You also could solve for $x$ instead and use that as your substitution! Using the first equation,



$$x = 108-y$$



or perhaps using our second equation



$$x = frac{156+3y}{2}$$



Any of these four substitutions is completely valid. These substitutions turn one of the equations into a true statement (notice how our original substitution turned the first equation, after simplifying, into $108=108$?). The other equation then becomes an equation of just one variable (after simplifying, the second equation in our original substitution becomes $-x-324=156$).



This also is a valid method if you have $n$ equations in $n$ variables, too, but it can become more complicated since you have to solve for all but one of the variables and substitute them into an equation. It can get messy pretty fast.






share|cite|improve this answer









$endgroup$





















    1












    $begingroup$

    If one of the numbers is $x$ and the sum of the two is 108, the other number is $108 - x$. If $x ge 108 - x$, we write



    $2x = 3(108 - x) + 156; tag 1$



    $2x = 324 - 3x + 156 = 480 - 3x; tag 2$



    $5x = 480; tag 3$



    $x = 96; tag 4$



    $108 - x = 12. tag 5$



    OR, if $108 - x ge x$,



    $2(108 - x) = 3x + 156; tag 6$



    $216 - 2x = 3x + 156; tag 7$



    $5x = 60; tag 8$



    $x = 12; tag 9$



    $108 - x = 96. tag{10}$






    share|cite|improve this answer









    $endgroup$













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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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      3












      $begingroup$

      So let's piece this together:



      "The sum of two numbers is $108$" - this gives us $x+y = 108$



      "the double of the greater exceeds the triple of the minor in 156"- Without loss of generality, we can assume $x>y$, and thus see $2x - 3y = 156$.



      Thus, we have a system of equations, which we seek to find $x$ and $y$:



      $$x+y = 108$$
      $$2x - 3y = 156$$



      You could solve this by whatever method you prefer to do so. Since you seek to use only one variable, notice that the first equation implies $y = 108-x$, yielding



      $$x+(108-x)= 108$$
      $$2x - 3(108-x) = 156$$



      This is a technique known as solving by substitution: solving for one of the variables and then substituting the resulting expression into the other equations.



      Notice, too, if you used the second of the equations from our original system, you could have found



      $$y = frac{156-2x}{-3}$$



      You also could solve for $x$ instead and use that as your substitution! Using the first equation,



      $$x = 108-y$$



      or perhaps using our second equation



      $$x = frac{156+3y}{2}$$



      Any of these four substitutions is completely valid. These substitutions turn one of the equations into a true statement (notice how our original substitution turned the first equation, after simplifying, into $108=108$?). The other equation then becomes an equation of just one variable (after simplifying, the second equation in our original substitution becomes $-x-324=156$).



      This also is a valid method if you have $n$ equations in $n$ variables, too, but it can become more complicated since you have to solve for all but one of the variables and substitute them into an equation. It can get messy pretty fast.






      share|cite|improve this answer









      $endgroup$


















        3












        $begingroup$

        So let's piece this together:



        "The sum of two numbers is $108$" - this gives us $x+y = 108$



        "the double of the greater exceeds the triple of the minor in 156"- Without loss of generality, we can assume $x>y$, and thus see $2x - 3y = 156$.



        Thus, we have a system of equations, which we seek to find $x$ and $y$:



        $$x+y = 108$$
        $$2x - 3y = 156$$



        You could solve this by whatever method you prefer to do so. Since you seek to use only one variable, notice that the first equation implies $y = 108-x$, yielding



        $$x+(108-x)= 108$$
        $$2x - 3(108-x) = 156$$



        This is a technique known as solving by substitution: solving for one of the variables and then substituting the resulting expression into the other equations.



        Notice, too, if you used the second of the equations from our original system, you could have found



        $$y = frac{156-2x}{-3}$$



        You also could solve for $x$ instead and use that as your substitution! Using the first equation,



        $$x = 108-y$$



        or perhaps using our second equation



        $$x = frac{156+3y}{2}$$



        Any of these four substitutions is completely valid. These substitutions turn one of the equations into a true statement (notice how our original substitution turned the first equation, after simplifying, into $108=108$?). The other equation then becomes an equation of just one variable (after simplifying, the second equation in our original substitution becomes $-x-324=156$).



        This also is a valid method if you have $n$ equations in $n$ variables, too, but it can become more complicated since you have to solve for all but one of the variables and substitute them into an equation. It can get messy pretty fast.






        share|cite|improve this answer









        $endgroup$
















          3












          3








          3





          $begingroup$

          So let's piece this together:



          "The sum of two numbers is $108$" - this gives us $x+y = 108$



          "the double of the greater exceeds the triple of the minor in 156"- Without loss of generality, we can assume $x>y$, and thus see $2x - 3y = 156$.



          Thus, we have a system of equations, which we seek to find $x$ and $y$:



          $$x+y = 108$$
          $$2x - 3y = 156$$



          You could solve this by whatever method you prefer to do so. Since you seek to use only one variable, notice that the first equation implies $y = 108-x$, yielding



          $$x+(108-x)= 108$$
          $$2x - 3(108-x) = 156$$



          This is a technique known as solving by substitution: solving for one of the variables and then substituting the resulting expression into the other equations.



          Notice, too, if you used the second of the equations from our original system, you could have found



          $$y = frac{156-2x}{-3}$$



          You also could solve for $x$ instead and use that as your substitution! Using the first equation,



          $$x = 108-y$$



          or perhaps using our second equation



          $$x = frac{156+3y}{2}$$



          Any of these four substitutions is completely valid. These substitutions turn one of the equations into a true statement (notice how our original substitution turned the first equation, after simplifying, into $108=108$?). The other equation then becomes an equation of just one variable (after simplifying, the second equation in our original substitution becomes $-x-324=156$).



          This also is a valid method if you have $n$ equations in $n$ variables, too, but it can become more complicated since you have to solve for all but one of the variables and substitute them into an equation. It can get messy pretty fast.






          share|cite|improve this answer









          $endgroup$



          So let's piece this together:



          "The sum of two numbers is $108$" - this gives us $x+y = 108$



          "the double of the greater exceeds the triple of the minor in 156"- Without loss of generality, we can assume $x>y$, and thus see $2x - 3y = 156$.



          Thus, we have a system of equations, which we seek to find $x$ and $y$:



          $$x+y = 108$$
          $$2x - 3y = 156$$



          You could solve this by whatever method you prefer to do so. Since you seek to use only one variable, notice that the first equation implies $y = 108-x$, yielding



          $$x+(108-x)= 108$$
          $$2x - 3(108-x) = 156$$



          This is a technique known as solving by substitution: solving for one of the variables and then substituting the resulting expression into the other equations.



          Notice, too, if you used the second of the equations from our original system, you could have found



          $$y = frac{156-2x}{-3}$$



          You also could solve for $x$ instead and use that as your substitution! Using the first equation,



          $$x = 108-y$$



          or perhaps using our second equation



          $$x = frac{156+3y}{2}$$



          Any of these four substitutions is completely valid. These substitutions turn one of the equations into a true statement (notice how our original substitution turned the first equation, after simplifying, into $108=108$?). The other equation then becomes an equation of just one variable (after simplifying, the second equation in our original substitution becomes $-x-324=156$).



          This also is a valid method if you have $n$ equations in $n$ variables, too, but it can become more complicated since you have to solve for all but one of the variables and substitute them into an equation. It can get messy pretty fast.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 16 '18 at 0:27









          Eevee TrainerEevee Trainer

          5,4691936




          5,4691936























              1












              $begingroup$

              If one of the numbers is $x$ and the sum of the two is 108, the other number is $108 - x$. If $x ge 108 - x$, we write



              $2x = 3(108 - x) + 156; tag 1$



              $2x = 324 - 3x + 156 = 480 - 3x; tag 2$



              $5x = 480; tag 3$



              $x = 96; tag 4$



              $108 - x = 12. tag 5$



              OR, if $108 - x ge x$,



              $2(108 - x) = 3x + 156; tag 6$



              $216 - 2x = 3x + 156; tag 7$



              $5x = 60; tag 8$



              $x = 12; tag 9$



              $108 - x = 96. tag{10}$






              share|cite|improve this answer









              $endgroup$


















                1












                $begingroup$

                If one of the numbers is $x$ and the sum of the two is 108, the other number is $108 - x$. If $x ge 108 - x$, we write



                $2x = 3(108 - x) + 156; tag 1$



                $2x = 324 - 3x + 156 = 480 - 3x; tag 2$



                $5x = 480; tag 3$



                $x = 96; tag 4$



                $108 - x = 12. tag 5$



                OR, if $108 - x ge x$,



                $2(108 - x) = 3x + 156; tag 6$



                $216 - 2x = 3x + 156; tag 7$



                $5x = 60; tag 8$



                $x = 12; tag 9$



                $108 - x = 96. tag{10}$






                share|cite|improve this answer









                $endgroup$
















                  1












                  1








                  1





                  $begingroup$

                  If one of the numbers is $x$ and the sum of the two is 108, the other number is $108 - x$. If $x ge 108 - x$, we write



                  $2x = 3(108 - x) + 156; tag 1$



                  $2x = 324 - 3x + 156 = 480 - 3x; tag 2$



                  $5x = 480; tag 3$



                  $x = 96; tag 4$



                  $108 - x = 12. tag 5$



                  OR, if $108 - x ge x$,



                  $2(108 - x) = 3x + 156; tag 6$



                  $216 - 2x = 3x + 156; tag 7$



                  $5x = 60; tag 8$



                  $x = 12; tag 9$



                  $108 - x = 96. tag{10}$






                  share|cite|improve this answer









                  $endgroup$



                  If one of the numbers is $x$ and the sum of the two is 108, the other number is $108 - x$. If $x ge 108 - x$, we write



                  $2x = 3(108 - x) + 156; tag 1$



                  $2x = 324 - 3x + 156 = 480 - 3x; tag 2$



                  $5x = 480; tag 3$



                  $x = 96; tag 4$



                  $108 - x = 12. tag 5$



                  OR, if $108 - x ge x$,



                  $2(108 - x) = 3x + 156; tag 6$



                  $216 - 2x = 3x + 156; tag 7$



                  $5x = 60; tag 8$



                  $x = 12; tag 9$



                  $108 - x = 96. tag{10}$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 16 '18 at 1:31









                  Robert LewisRobert Lewis

                  44.6k22964




                  44.6k22964






























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                      .mw-parser-output .infobox{border:1px solid #aaa;background-color:#f9f9f9;color:black;margin:.5em 0 .5em 1em;padding:.2em;float:right;clear:right;width:22em;text-align:left;font-size:88%;line-height:1.6em}.mw-parser-output .infobox td,.mw-parser-output .infobox th{vertical-align:top;padding:0 .2em}.mw-parser-output .infobox caption{font-size:larger}.mw-parser-output .infobox.bordered{border-collapse:collapse}.mw-parser-output .infobox.bordered td,.mw-parser-output .infobox.bordered th{border:1px solid #aaa}.mw-parser-output .infobox.bordered .borderless td,.mw-parser-output .infobox.bordered .borderless th{border:0}.mw-parser-output .infobox-showbutton .mw-collapsible-text{color:inherit}.mw-parser-output .infobox.bordered .mergedtoprow td,.mw-parser-output .infobox.bordered .mergedtoprow th{border:0;border-top:1px solid #aaa;border-right:1px solid #aaa}.mw-parser-output .infobox.bordered .mergedrow td,.mw-parser-output .infobox.bordered .mergedrow th{border:0;border-right:1px solid
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