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Easier way to solve a LES (only pen/paper, no calculator)

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0

$begingroup$

I want to solve the following Linear Equation System with only pen and paper;

$$
470 = x_A - frac310x_B tag1
$$
$$
940 = x_B - frac210x_A tag2
$$

I attempt to solve for $x_A$ by inserting equation (2) into (1) and rewriting it, etc. I got something like;

$$
x_A=470 +frac310(940+frac210x_A) Rightarrow dots Rightarrow x_Afrac94100=470+frac310940=470+frac282010
$$

Next I try to isolate $x_A$ and simplify the expression;

$$
dots Rightarrow x_A =frac10094(470+frac282010)=frac10094(470+282)=frac10094752
$$

This is where I get stuck. Have I dug myself into a difficult hole when there is a better/more efficient approach to solving this LES? Or am I just missing a good method to solve $x_A=frac10094cdot752$ to get it to 800?

Any insight into "how to think" here and how you solve it with just pen and paper is much appreciated.

(Solution: $x_A=800$, $x_B=1100$)

linear-algebra systems-of-equations problem-solving

share|cite|improve this question

edited Mar 21 at 11:06

litmus

asked Mar 21 at 10:02

litmuslitmus

297318

$endgroup$

add a comment | 

0

$begingroup$

I want to solve the following Linear Equation System with only pen and paper;

$$
470 = x_A - frac310x_B tag1
$$
$$
940 = x_B - frac210x_A tag2
$$

I attempt to solve for $x_A$ by inserting equation (2) into (1) and rewriting it, etc. I got something like;

$$
x_A=470 +frac310(940+frac210x_A) Rightarrow dots Rightarrow x_Afrac94100=470+frac310940=470+frac282010
$$

Next I try to isolate $x_A$ and simplify the expression;

$$
dots Rightarrow x_A =frac10094(470+frac282010)=frac10094(470+282)=frac10094752
$$

This is where I get stuck. Have I dug myself into a difficult hole when there is a better/more efficient approach to solving this LES? Or am I just missing a good method to solve $x_A=frac10094cdot752$ to get it to 800?

Any insight into "how to think" here and how you solve it with just pen and paper is much appreciated.

(Solution: $x_A=800$, $x_B=1100$)

linear-algebra systems-of-equations problem-solving

share|cite|improve this question

edited Mar 21 at 11:06

litmus

asked Mar 21 at 10:02

litmuslitmus

297318

$endgroup$

add a comment | 

$begingroup$

I want to solve the following Linear Equation System with only pen and paper;

$$
470 = x_A - frac310x_B tag1
$$
$$
940 = x_B - frac210x_A tag2
$$

I attempt to solve for $x_A$ by inserting equation (2) into (1) and rewriting it, etc. I got something like;

$$
x_A=470 +frac310(940+frac210x_A) Rightarrow dots Rightarrow x_Afrac94100=470+frac310940=470+frac282010
$$

Next I try to isolate $x_A$ and simplify the expression;

$$
dots Rightarrow x_A =frac10094(470+frac282010)=frac10094(470+282)=frac10094752
$$

This is where I get stuck. Have I dug myself into a difficult hole when there is a better/more efficient approach to solving this LES? Or am I just missing a good method to solve $x_A=frac10094cdot752$ to get it to 800?

Any insight into "how to think" here and how you solve it with just pen and paper is much appreciated.

(Solution: $x_A=800$, $x_B=1100$)

linear-algebra systems-of-equations problem-solving

share|cite|improve this question

edited Mar 21 at 11:06

litmus

asked Mar 21 at 10:02

litmuslitmus

297318

$endgroup$

share|cite|improve this question

edited Mar 21 at 11:06

litmus

asked Mar 21 at 10:02

litmuslitmus

297318

share|cite|improve this question

edited Mar 21 at 11:06

litmus

asked Mar 21 at 10:02

litmuslitmus

297318

asked Mar 21 at 10:02

litmuslitmus

297318

asked Mar 21 at 10:02

litmuslitmus

297318

3 Answers
3

active

oldest

votes

1

$begingroup$

First notice $94=47cdot 2$. You can simplify before adding/multiplying:
$$x_Afrac94100=470+frac310940colorred=470+frac282010 Rightarrow \
x_Afrac47cdot 2100=47cdot 10+frac310cdot 47cdot 20 Rightarrow \
x_Afrac2100=10+frac6010 Rightarrow \
2x_A=1000+600 Rightarrow \
x_A=800.$$

share|cite|improve this answer

answered Mar 21 at 10:31

farruhotafarruhota

21.8k2842

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, but why do we focus on factorizing 94 into 47 and 2? Is it because we have 940 and 470 on the RHS?
  $endgroup$
  – litmus
  Mar 21 at 10:34
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  yes, look for common factors
  $endgroup$
  – farruhota
  Mar 21 at 10:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for your help. Wouldn't it be more intuitive if the red text is $470+frac310cdot470cdot2$?
  $endgroup$
  – litmus
  Mar 21 at 10:45
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  the first line is yours and the red text implies it must be deleted, only black text is the right solution. in other words, you should not write this red text (it is an error)
  $endgroup$
  – farruhota
  Mar 21 at 10:48
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  you are welcome. the more problems one solves, the more trained the eyes get to notice the connections between parts.
  $endgroup$
  – farruhota
  Mar 21 at 10:50
  

  
  
  
  

 | 
show 1 more comment

1

$begingroup$

May be, it could be simpler to do $$470 = x_A - frac310x_B implies4700=10x_A-3x_Btag1$$

$$940 = x_B - frac210x_A implies9400=10x_B-2x_Atag2$$
Multiply $(1)$ by $10$ and $(2)$ by $3$ and add them together; this gives immediately $$75200=94 x_Aimplies x_A=frac7520094=800$$ Plug this in $(2)$ to get $x_B$

share|cite|improve this answer

answered Mar 21 at 10:23

Claude LeiboviciClaude Leibovici

125k1158135

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! But with pen and paper, how do you compute 75200/94 ?
  $endgroup$
  – litmus
  Mar 21 at 10:25
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @litmus. By hand !!
  $endgroup$
  – Claude Leibovici
  Mar 21 at 10:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you please tell me the name of the method used for this so I can google it.
  $endgroup$
  – litmus
  Mar 21 at 10:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @litmus. Have a look at tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx
  $endgroup$
  – Claude Leibovici
  Mar 21 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But this link explains how to solve Linear Systems, I would like to know how you solve 75200/94 by hand. Please show me, this would help me so much!
  $endgroup$
  – litmus
  Mar 21 at 10:41
  

  
  
  
  

 | 
show 2 more comments

0

$begingroup$

What are your problems ?

$ x_A=frac10094cdot752$ is correct, since $752:94=8.$

share|cite|improve this answer

answered Mar 21 at 10:07

FredFred

48.6k11849

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But how do you see that 752 divided by 94 is 8? I.e. how do you get to 800?
  $endgroup$
  – litmus
  Mar 21 at 10:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $90 times 8=720$ and $4 times 8=32.$
  $endgroup$
  – Fred
  Mar 21 at 10:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you please elaborate, do you use 90 and 4 because they sum to 94?
  $endgroup$
  – litmus
  Mar 21 at 10:18
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, we have 90+4=94
  $endgroup$
  – Fred
  Mar 21 at 10:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for your help! But how do you know to decompose 752 into 720 and 32? (If you don't know that $x_A=800$)
  $endgroup$
  – litmus
  Mar 21 at 10:23
  

  
  
  
  

add a comment | 

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1

$begingroup$

First notice $94=47cdot 2$. You can simplify before adding/multiplying:
$$x_Afrac94100=470+frac310940colorred=470+frac282010 Rightarrow \
x_Afrac47cdot 2100=47cdot 10+frac310cdot 47cdot 20 Rightarrow \
x_Afrac2100=10+frac6010 Rightarrow \
2x_A=1000+600 Rightarrow \
x_A=800.$$

share|cite|improve this answer

answered Mar 21 at 10:31

farruhotafarruhota

21.8k2842

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, but why do we focus on factorizing 94 into 47 and 2? Is it because we have 940 and 470 on the RHS?
  $endgroup$
  – litmus
  Mar 21 at 10:34
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  yes, look for common factors
  $endgroup$
  – farruhota
  Mar 21 at 10:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for your help. Wouldn't it be more intuitive if the red text is $470+frac310cdot470cdot2$?
  $endgroup$
  – litmus
  Mar 21 at 10:45
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  the first line is yours and the red text implies it must be deleted, only black text is the right solution. in other words, you should not write this red text (it is an error)
  $endgroup$
  – farruhota
  Mar 21 at 10:48
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  you are welcome. the more problems one solves, the more trained the eyes get to notice the connections between parts.
  $endgroup$
  – farruhota
  Mar 21 at 10:50
  

  
  
  
  

 | 
show 1 more comment

1

$begingroup$

First notice $94=47cdot 2$. You can simplify before adding/multiplying:
$$x_Afrac94100=470+frac310940colorred=470+frac282010 Rightarrow \
x_Afrac47cdot 2100=47cdot 10+frac310cdot 47cdot 20 Rightarrow \
x_Afrac2100=10+frac6010 Rightarrow \
2x_A=1000+600 Rightarrow \
x_A=800.$$

share|cite|improve this answer

answered Mar 21 at 10:31

farruhotafarruhota

21.8k2842

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Ok, but why do we focus on factorizing 94 into 47 and 2? Is it because we have 940 and 470 on the RHS?
  $endgroup$
  – litmus
  Mar 21 at 10:34
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  yes, look for common factors
  $endgroup$
  – farruhota
  Mar 21 at 10:35
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for your help. Wouldn't it be more intuitive if the red text is $470+frac310cdot470cdot2$?
  $endgroup$
  – litmus
  Mar 21 at 10:45
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  the first line is yours and the red text implies it must be deleted, only black text is the right solution. in other words, you should not write this red text (it is an error)
  $endgroup$
  – farruhota
  Mar 21 at 10:48
  
  

  
  
  
  


• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  you are welcome. the more problems one solves, the more trained the eyes get to notice the connections between parts.
  $endgroup$
  – farruhota
  Mar 21 at 10:50
  

  
  
  
  

 | 
show 1 more comment

$begingroup$

First notice $94=47cdot 2$. You can simplify before adding/multiplying:
$$x_Afrac94100=470+frac310940colorred=470+frac282010 Rightarrow \
x_Afrac47cdot 2100=47cdot 10+frac310cdot 47cdot 20 Rightarrow \
x_Afrac2100=10+frac6010 Rightarrow \
2x_A=1000+600 Rightarrow \
x_A=800.$$

share|cite|improve this answer

answered Mar 21 at 10:31

farruhotafarruhota

21.8k2842

$endgroup$

share|cite|improve this answer

answered Mar 21 at 10:31

farruhotafarruhota

21.8k2842

answered Mar 21 at 10:31

farruhotafarruhota

21.8k2842

answered Mar 21 at 10:31

farruhotafarruhota

21.8k2842

1

$begingroup$

May be, it could be simpler to do $$470 = x_A - frac310x_B implies4700=10x_A-3x_Btag1$$

$$940 = x_B - frac210x_A implies9400=10x_B-2x_Atag2$$
Multiply $(1)$ by $10$ and $(2)$ by $3$ and add them together; this gives immediately $$75200=94 x_Aimplies x_A=frac7520094=800$$ Plug this in $(2)$ to get $x_B$

share|cite|improve this answer

answered Mar 21 at 10:23

Claude LeiboviciClaude Leibovici

125k1158135

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! But with pen and paper, how do you compute 75200/94 ?
  $endgroup$
  – litmus
  Mar 21 at 10:25
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @litmus. By hand !!
  $endgroup$
  – Claude Leibovici
  Mar 21 at 10:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you please tell me the name of the method used for this so I can google it.
  $endgroup$
  – litmus
  Mar 21 at 10:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @litmus. Have a look at tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx
  $endgroup$
  – Claude Leibovici
  Mar 21 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But this link explains how to solve Linear Systems, I would like to know how you solve 75200/94 by hand. Please show me, this would help me so much!
  $endgroup$
  – litmus
  Mar 21 at 10:41
  

  
  
  
  

 | 
show 2 more comments

1

$begingroup$

May be, it could be simpler to do $$470 = x_A - frac310x_B implies4700=10x_A-3x_Btag1$$

$$940 = x_B - frac210x_A implies9400=10x_B-2x_Atag2$$
Multiply $(1)$ by $10$ and $(2)$ by $3$ and add them together; this gives immediately $$75200=94 x_Aimplies x_A=frac7520094=800$$ Plug this in $(2)$ to get $x_B$

share|cite|improve this answer

answered Mar 21 at 10:23

Claude LeiboviciClaude Leibovici

125k1158135

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks! But with pen and paper, how do you compute 75200/94 ?
  $endgroup$
  – litmus
  Mar 21 at 10:25
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @litmus. By hand !!
  $endgroup$
  – Claude Leibovici
  Mar 21 at 10:27
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you please tell me the name of the method used for this so I can google it.
  $endgroup$
  – litmus
  Mar 21 at 10:29
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @litmus. Have a look at tutorial.math.lamar.edu/Classes/Alg/SystemsTwoVrble.aspx
  $endgroup$
  – Claude Leibovici
  Mar 21 at 10:39
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But this link explains how to solve Linear Systems, I would like to know how you solve 75200/94 by hand. Please show me, this would help me so much!
  $endgroup$
  – litmus
  Mar 21 at 10:41
  

  
  
  
  

 | 
show 2 more comments

$begingroup$

May be, it could be simpler to do $$470 = x_A - frac310x_B implies4700=10x_A-3x_Btag1$$

$$940 = x_B - frac210x_A implies9400=10x_B-2x_Atag2$$
Multiply $(1)$ by $10$ and $(2)$ by $3$ and add them together; this gives immediately $$75200=94 x_Aimplies x_A=frac7520094=800$$ Plug this in $(2)$ to get $x_B$

share|cite|improve this answer

answered Mar 21 at 10:23

Claude LeiboviciClaude Leibovici

125k1158135

$endgroup$

share|cite|improve this answer

answered Mar 21 at 10:23

Claude LeiboviciClaude Leibovici

125k1158135

answered Mar 21 at 10:23

Claude LeiboviciClaude Leibovici

125k1158135

answered Mar 21 at 10:23

Claude LeiboviciClaude Leibovici

125k1158135

0

$begingroup$

What are your problems ?

$ x_A=frac10094cdot752$ is correct, since $752:94=8.$

share|cite|improve this answer

answered Mar 21 at 10:07

FredFred

48.6k11849

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  But how do you see that 752 divided by 94 is 8? I.e. how do you get to 800?
  $endgroup$
  – litmus
  Mar 21 at 10:10
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  $90 times 8=720$ and $4 times 8=32.$
  $endgroup$
  – Fred
  Mar 21 at 10:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Could you please elaborate, do you use 90 and 4 because they sum to 94?
  $endgroup$
  – litmus
  Mar 21 at 10:18
  
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Yes, we have 90+4=94
  $endgroup$
  – Fred
  Mar 21 at 10:21
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thanks for your help! But how do you know to decompose 752 into 720 and 32? (If you don't know that $x_A=800$)
  $endgroup$
  – litmus
  Mar 21 at 10:23
  

  
  
  
  

add a comment | 

0

$begingroup$

What are your problems ?

$ x_A=frac10094cdot752$ is correct, since $752:94=8.$

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answered Mar 21 at 10:07

FredFred

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  But how do you see that 752 divided by 94 is 8? I.e. how do you get to 800?
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  – litmus
  Mar 21 at 10:10
  
  

  
  
  
  


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  $90 times 8=720$ and $4 times 8=32.$
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  – Fred
  Mar 21 at 10:13
  

  
  
  
  


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  Could you please elaborate, do you use 90 and 4 because they sum to 94?
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  – litmus
  Mar 21 at 10:18
  
  

  
  
  
  


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  Yes, we have 90+4=94
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  – Fred
  Mar 21 at 10:21
  

  
  
  
  


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  Thanks for your help! But how do you know to decompose 752 into 720 and 32? (If you don't know that $x_A=800$)
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  – litmus
  Mar 21 at 10:23
  

  
  
  
  

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What are your problems ?

$ x_A=frac10094cdot752$ is correct, since $752:94=8.$

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answered Mar 21 at 10:07

FredFred

48.6k11849

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share|cite|improve this answer

answered Mar 21 at 10:07

FredFred

48.6k11849

answered Mar 21 at 10:07

FredFred

48.6k11849

answered Mar 21 at 10:07

FredFred

48.6k11849