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How to solve simultaneous linear equations … using algebra

February 15, 2013 by Simon 114 Comments

Download a copy of the questions here: QT Simultaneous Linear Equations

Download a copy of the answers: QT Simultaneous Linear Equations ANSWERS

These videos show how to solve simultaneous linear equations in steps.

  • The first video should be relatively straightforward as it only deals with positive numbers.
  • The second is a little trickier (around level 5) and involves dealing with a negative term.
  • The third video … shows more of a real application..

 


Download a copy of the questions here: QT Simultaneous Linear Equations

Download a copy of the answers: QT Simultaneous Equations ANSWERS

Learning how to solve simultaneous linear equations can be important for applications in economics, such as working out the best price to sell a product. This is usually called ‘supply and demand.’

Imagine you make pencils:

  • If you sell at a high price they’ll be less demand
  • If you sell at a low price they’ll be too many and less profit

Simultaneous equations can be created to show how quantities sold vary with supply and demand. These can then be solved to show the best price to make sure you sell your pencils, and the demand continues.

Another example – a favourite in exams – is to use mobile phone contracts. Sometimes these are  given as a graph and there’s more about this in the next post. Although the question is usually two linear equations, and it asks you to pick the best value.

One of my favourite exam questions involves The Khans and The Smiths buying theatre tickets. Each family has got different numbers of adults and children … and you need to create a couple of simultaneous equations to work out the price of each ticket.

These kind of questions may be a little strange (why didn’t they just ring the box office?), but they do give an insight into how equations work. There are other examples such as arranging a meeting half way through a journey or working out the cost of  bank loans.

Please add a comment below with any more real life examples.

Watch the videos on YouTube:

How to solve simultaneous linear equations using algebra 

How to solve simultaneous linear equations using algebra 

Simultaneous equations GCSE higher maths 

Simultaneous Equation Word Problems – GCSE maths higher tier 

How to solve simultaneous equation word problems 

Solve a Simultaneous Linear Equation – word problem QV Q7 

Solve a Simultaneous Linear Equation – word problem QV Q8 

Solve a Simultaneous Linear Equation – word problem QV Q9 

Solve a Simultaneous Linear Equation – word problem QV Q10 

 

Comments

  1. Georgie says

    September 18, 2015 at 5:26 pm

    r+s/8s = r-s/8n

    I have this equation to rearrange to make r the subject (apologies, but don’t have a divide button on my computer!!!!!!)

    I’m really confused so would appreciate any feedback you have that could help me

    Many thanks

    Reply
    • Simon says

      September 20, 2015 at 11:23 am

      Hi Georgie

      I’ve been working through this but please could you check the question? The couple of solutions I have cancel out the r (as it’s just one value of r each side).

      Reply
  2. Renea Chua says

    June 30, 2015 at 1:16 pm

    How do you do this:
    a – 5b = -12
    8a + 3b = -10

    Reply
  3. MagicMinter HD says

    June 24, 2015 at 5:21 pm

    wtf i did the same method with this question and didnt get it right

    4x+y=25
    x-3y=16

    Reply
  4. InsaneGamer says

    June 7, 2015 at 3:03 pm

    where have you been all this time??

    Reply
  5. SuperSonicSpeedGuy says

    June 3, 2015 at 7:02 pm

    The camera keeps going all blurry, like at 4:00 🙁 But thanks for the vid!

    Reply
  6. Hannah Dale says

    June 3, 2015 at 5:39 pm

    Thank you so much for these videos! I’m currently revising for end of year
    exams and this is helping alot! Very much appreciated.

    Reply
  7. sandev randhawa says

    June 3, 2015 at 3:07 pm

    great video to watch

    Reply
  8. Ajit King says

    June 1, 2015 at 3:17 pm

    did you know science

    Reply
  9. Ajit King says

    June 1, 2015 at 3:06 pm

    you are a very good teacher dude.

    Reply
  10. CoolCat4196 says

    May 17, 2015 at 7:03 am

    I have this equation that I would like you to help me with, if possible.

    Parking Station A: charges $3.20 (first hour) and $2.00 (afterwards)
    Parking Station B: charges $6.00 (first hour) and $1.20 (afterwards)

    Calculate the times of parking which would result in the cost being the
    same at both parking stations.

    I hope that you can help, Thanks and you are great

    Reply
  11. kurtis sankey says

    May 15, 2015 at 11:28 am

    having trouble with this one A television repair person charges a service
    call of $36 and then a further $28 per hour. An electrician charges a
    service call of $24 and then a further $36 per hour. Use simultaneous
    equations to calculate the number of hours each person worked to charge the
    same total fee. What is the fee?

    Reply
  12. Ahmed El-Aide says

    May 7, 2015 at 3:53 pm

    there is a question I found and I couldn’t solve it please help me
    the cost oh hire a tent consists of two parts
    $C and $D per day ( there is a plus sign in the place of “and”)

    total cost for 4 days is $27.1 and for 7 days $34.30
    write down the 2 equations

    Reply
  13. J. Moore says

    May 7, 2015 at 2:32 pm

    THANK YOU SO MUCH YOU HAVE SAVED MY LIFE

    Reply
  14. unknown profile says

    May 6, 2015 at 7:28 pm

    a and b are positive whole numbers.
    find value of a and b to make the solution to this equation x=4
    a(x+1)+b(x-2)=16 please help me if you can 🙂

    Reply
  15. Lee Dodd says

    May 4, 2015 at 4:37 pm

    Do you have any videos on Algebra and area.I am not sure what the correct
    name for it is, EG A triangle is drawn with 3x-15 then 2x-24 then one side
    saying x+2.Then you have to work out the Perimeter.

    Reply
  16. The Astute Atheist. says

    May 3, 2015 at 4:57 am

    Hey, it may be a bit late, as the video was uploaded two years ago
    (almost). But I have a question, and I have NO clue how to do it. It’s a
    big one with a table so I will try my best:

    *Question:*
    A company makes three types of patio furniture, chairs, rockers and chaise
    lounges. Each require wood, plastic and aluminium, as given by the table
    below. The company has 400 units of wood, 600 units of plastic and 1500
    units of aluminium.

    *Table:*
    Wood Plastic Aluminium
    Chair 1 Unit 1 Unit 2 Units
    Rocker 1 Unit 1 Unit 3 Units
    Chaise Lounge 1 Unit 2 Units 5 Units

    For it’s end of season production run, the company wants to use up ALL the
    stock. To do this, *calculate* how many chairs, rockers and chaise lounges
    should the company make?

    *We have been doing simultaneous equations and linear functions in class,
    that could help*. I really need help so, it’d be REALLY awesome if you
    could help me out.

    Reply
  17. ABDUL JAWAD Ghalib says

    April 16, 2015 at 5:21 am

    I hav given a symultaneous equation word problem it would be a grateful if
    you kindly give the solution of the following question:
    At a shooting range, each shot costs 20c. If you hit the target, you
    receive 30c. Mira has 20 shots and makes a loss of 70c. How many hits did
    she get?

    Reply
  18. On The Road Media says

    April 13, 2015 at 10:17 am

    “Solve algebraically these simultaneous equations.”
    y= 4x² – 9x -1
    y= 5-4x

    Reply
  19. willoxens says

    April 10, 2015 at 2:16 pm

    Could you help me with a question?
    It asks to consider the equations:
    (k-3)x+5y=-2
    2x-3y=4
    where k is a real constant for which I have to find the unique solution.
    Thanks in advance!

    Reply
  20. Ambra Onyss says

    April 6, 2015 at 10:27 pm

    where did x=4 come from?

    Reply
  21. yassin ramadan says

    April 6, 2015 at 12:30 pm

    thanx a lot you really did help with this. im going to do my GCSE after a
    month so thanx again

    Reply
  22. hi ih says

    March 31, 2015 at 6:46 pm

    how is it x=4? where did this come from? it got me so confused. pls reply

    Reply
  23. CoD Shotz says

    March 21, 2015 at 3:04 pm

    Thank you this was really helpful.

    Reply
  24. stephen barker says

    March 2, 2015 at 12:15 pm

    q1 there are 1130 pupils in a school and no classes have more than 32
    pupils. how many classrooms could be used show this information as an
    inequality. q2 a person is prepared to spend £300 taking friends
    out to celebrate. if the restaurant charges £12 per head how many guests
    can be invited show this information as an inequality can you help please

    Reply
  25. Shoot Fresh Mine says

    February 18, 2015 at 12:17 pm

    hey can u help me answer these questions

    -5m-3n=-3
    10m-7n=-12

    AND

    -2v+w=-19
    -2v-7=21

    thanks

    Reply
  26. Shoot Fresh Mine says

    February 11, 2015 at 6:05 pm

    Hey I got a question which I need help with working out.

    3u+v=17
    7u+v=29

    Reply
  27. Dafydd Morris says

    February 9, 2015 at 7:17 pm

    How would I do
    4x+3Y=12
    6x+2Y=13

    Reply
  28. Dafydd Morris says

    February 9, 2015 at 7:04 pm

    Praise be Simon deacon 😀

    Reply
  29. Shoot Fresh Mine says

    February 8, 2015 at 12:17 pm

    How would u work out

    3p+5q=43
    8p+5q=73

    Reply
  30. Cethical says

    February 7, 2015 at 6:34 pm

    You’re such a humble bloke. I miss London just because of your manners.

    Reply
  31. ABWEndon says

    February 6, 2015 at 8:33 pm

    Thank you Simon. Very well explained and although my brain hurts a little,
    I can feel that it has done me some good!

    Reply
  32. vani na says

    February 3, 2015 at 10:26 am

    thank you so much

    Reply
  33. Phase Gaming says

    January 28, 2015 at 5:09 pm

    thanks i have a maths test coming up and you relay helped

    Reply
  34. stephen barker says

    January 24, 2015 at 7:20 pm

    have i got this right if both equations are + you subtract if both
    minus you add and 1 of each you add

    Reply
  35. Mathew Stenton says

    January 22, 2015 at 3:32 pm

    Wow Simon your my best friend and I love you thanks

    Reply
  36. OpiumMessiah says

    January 14, 2015 at 4:30 pm

    Thanks, man. 🙂

    Reply
  37. Phoenix Feathers says

    January 13, 2015 at 10:33 pm

    Thank You, Thank You, Thank You!!!!!!

    Reply
  38. Charlie Michele says

    January 13, 2015 at 12:00 am

    Thank you so much! I finally get it!

    Reply
  39. Daniel Gadd says

    January 6, 2015 at 3:34 am

    2y = 5x + 7
    3y = -2x -5

    Help me here someone please

    Reply
  40. Daniel Kibby says

    January 3, 2015 at 10:15 pm

    I’ve watched 3 of your videos as I am retaking my Maths GCSE and found them
    very helpful. Thank you.

    Reply
  41. Walter Payton says

    December 17, 2014 at 10:28 pm

    it helped me ooooooooooooooooooooooooooooooooooo!!!! may god bless you sir

    Reply
  42. MegaSaya64 says

    November 8, 2014 at 5:03 pm

    thankkk

    Reply
  43. Hayden Nicholson says

    November 6, 2014 at 5:45 pm

    Thanks +Simon Deacon helped alot!

    Reply
  44. bubbles xo says

    November 2, 2014 at 8:48 pm

    very helpful cheers

    Reply
  45. Marcin Szostok says

    October 27, 2014 at 7:34 pm

    THANK YOU !

    Reply
  46. thejchan16 says

    October 27, 2014 at 10:31 am

    Thanks for the video! It was so helpful leading up to my Maths Test. You
    are such a great teacher! Keep doing what you do because your awesome!

    Reply
  47. Michelle Wanha says

    October 20, 2014 at 6:35 pm

    Thank you sir!

    Reply
  48. Mateen Malik says

    October 19, 2014 at 2:52 pm

    Thanks makes a lot more sense.

    Reply
  49. Jose Dominguez says

    September 30, 2014 at 11:35 pm

    Fix the audio

    Reply
  50. ibraheem nawaz says

    September 28, 2014 at 8:49 pm

    Awesome Best tutor you are better than my maths teacher.. 🙂

    Reply
  51. Tendaishe Mbele says

    September 9, 2014 at 6:59 pm

    thank you so much, i was confused before i watched this video 😀

    Reply
  52. stephen barker says

    September 2, 2014 at 1:20 pm

    im trying understand something quadratic equations no 1 x sqaured
    minus x minus 56 =0 changed from minus b at start to positive minus minus
    is a plus comes out right as 8 and minus 7 next question 3x sqaured +
    7x minus 13 =0 but with this one if you change the minus at start to plus
    answers in back of book are different to what it comes up with any advice.

    Reply
  53. Matthew Sydoruk says

    August 7, 2014 at 6:07 am

    hi

    Reply
  54. stephen barker says

    August 6, 2014 at 10:32 am

    can you check some equations been told there wrong 4x plus 2y = 12 x +
    2y = 6 2y cancels out 4x minus x = 3x 3x = 12 minus 6 is 6 so x = 2 2
    plus 2y = 6 y = 2 so x is 2 and y is 2 been told can only use each
    number once

    Reply
  55. yukicross6767 says

    July 20, 2014 at 6:09 am

    you see i have this equation tht i dont get may u help me?
    see i have {5x+2y=19 … 1
    {2x+3y=12 …2
    but i dont get how you r suppose to multiple this by 3 for the top and then
    u need to multiple by 2 for the second part

    Reply
  56. Peter Backstrand says

    July 14, 2014 at 12:48 pm

    how became x=4 all of a sudden? 7:45

    Reply
  57. stephen barker says

    July 3, 2014 at 11:10 am

    how do you solve simultanious equations graphically

    Reply
  58. Jibrael Malik says

    June 8, 2014 at 7:25 pm

    Thanks helped me out!

    Reply
  59. stephen barker says

    June 6, 2014 at 1:33 pm

    i do simultanious equations using the same method as in the vid

    Reply
  60. HilalFener says

    June 4, 2014 at 11:51 pm

    Why is y=2? im kind of confused how you got 2

    Reply
  61. stephen barker says

    June 3, 2014 at 2:25 pm

    got it. if a farmer wishes to promote the white ridge back sow as the most
    prolific breeder then which of the 3averages would he not include

    Reply
  62. stephen barker says

    June 3, 2014 at 2:13 pm

    youll have do some with minus terms

    Reply
  63. stephen barker says

    May 26, 2014 at 7:45 pm

    how would i solve x squared + y squared = 29
    and y _ x = 3

    Reply
  64. Josh Shil says

    May 25, 2014 at 12:43 pm

    Hi Simon Deacon I am looking for questions that give the answer of both x’s
    and both y’s FOR SOLVING SIMULTANEOUS LINEAR EQUATIONS. Thank you

    Reply
  65. stephen barker says

    May 22, 2014 at 11:54 am

    x squared +y squared = 29 y minus x = 3 thats the first set set 2 is y+ 1
    =x squared and x = y minus 1

    Reply
  66. Cameron Richmond says

    May 20, 2014 at 12:04 am

    Very useful doing this in maths and I didn’t understand it till now and I
    can solve these in my up and coming maths exam

    Reply
  67. davidymcmb says

    May 12, 2014 at 5:54 am

    Thanks for the help man!

    Reply
  68. stephen barker says

    April 23, 2014 at 2:37 pm

    ive nailed equations with minus numbers already for next year now

    Reply
  69. BAT TT says

    March 30, 2014 at 8:45 am

    This is very helpful thank you

    Reply
  70. Justin Beary says

    March 23, 2014 at 2:44 pm

    Thank you so much I finally understand how to do it without making stupid
    mistakes wuhuuuuu!

    Reply
  71. Annie Lim says

    March 8, 2014 at 3:38 am

    it is really blur

    Reply
  72. mussa sherif says

    February 18, 2014 at 8:57 am

    Thanx

    Reply
  73. killa killa says

    February 10, 2014 at 9:25 pm

    Thanks you really helped I have a test tomorrow-hope I do well

    Reply
  74. stephen barker says

    January 29, 2014 at 12:55 pm

    i make 1 letter the same in both equations subtracted answers gets
    remaining letter

    Reply
  75. mohamed abdelhalim says

    January 12, 2014 at 8:45 am

    thanks !!! really did help.could you solve this question for me please. A
    submarine can travel at 25 knots with the current and at 16 knots against
    it .Find the speed of the wind and the speed of the submarine in still
    water.

    Reply
  76. SkizaKiza says

    December 9, 2013 at 9:45 pm

    thankyou for the help!

    Reply
  77. Simon Deacon says

    December 8, 2013 at 10:41 am

    Please like and leave a comment!

    Visit https://www.3minutemaths.co.uk for real maths, tips and techniques.

    Reply
  78. Satellite Dave says

    December 6, 2013 at 12:40 am

    Thank you very much!! Your video helped me a lot!

    Reply
  79. Finlay McKee says

    December 4, 2013 at 5:43 pm

    I have my GCSE’s this year and you have managed to teach me what my maths
    teacher has failed to. Thank you so much!

    Reply
  80. Anita Cheung says

    November 24, 2013 at 8:19 am

    thanks

    Reply
  81. Simon Deacon says

    November 6, 2013 at 9:10 am

    Hi – you need to plot both lines and see where they cross. It’s OK if you
    have an idea where they are likely to be on a graph… but it can take a
    long time to get that information. I’ll post a video on this and let you
    know. Simultaneous solving by using algebra is better and easier. All best S

    Reply
  82. stephen barker says

    November 6, 2013 at 12:18 am

    a graph how would i do that

    Reply
  83. Simon Deacon says

    November 5, 2013 at 9:51 pm

    Hi – this is really the easiest way. You could solve by plotting a graph
    but it takes a while and isn’t always very accurate. Keep practicing and
    good luck!

    Reply
  84. stephen barker says

    November 5, 2013 at 3:20 pm

    whats the easyest way to solve these

    Reply
  85. Simon Deacon says

    September 21, 2013 at 1:19 pm

    Hi – Q2 needs a bit more explanation. If you email me through Maths Wrap I’ll send a solution. In the meantime Q1:

    You’ve got 2 equations J = 2L and J + L = 5L – 48.

    So, change both and you’ll get J-2L = 0 and J-4L = -48. Then take eq2 from eq1 and you should get -2L = -48. So L = 24.

    Put L=24 back into eq1 and J = 48.

    So Jan is 48 and Lisa is 24.

    I hope this helps and all best S

    Reply
  86. mymindydiditong says

    September 20, 2013 at 11:07 pm

    *two

    Reply
  87. mymindydiditong says

    September 20, 2013 at 11:07 pm

    Hi, im not sure how to solve these teo problems:
    1)Jan is twice as old as Lisa. The sum of their ages is 5 times Lisa’s age minus 48. How old are they now?
    2)John received changes worth $13 all in coins. He received 10 more dimes than nickles, and 22 more quarters than dimes. How many coins of each did he receive?

    Reply
  88. Simon Deacon says

    September 19, 2013 at 11:35 am

    Hi Mustanser – glad you liked the video and thanks for the comment 🙂
    The first equation is F = 3S . The second is a little more difficult. Imagine 10 years ago … at that stage the father would be F – 10 and the son would be S – 10.
    However the dad is 5 times older so F – 10 = 5 (S – 10). Now you’ve got two equations F = 3S (or F – 3S = 0) and F – 10 = 5S – 50 (or F – 5S = -40). Take equation 2. away from equation 1. You should end up with the son aged 20 and the dad aged 60. All best S

    Reply
  89. mustanser hussain says

    September 18, 2013 at 6:52 pm

    ago*

    Reply
  90. mustanser hussain says

    September 18, 2013 at 6:15 pm

    Hey,this was very useful indeed,thankyou.Can u give me the solution for this problem. A man is 3 times the age of his son.10 years aga he was five times the age of his son.Find their ages by finding the value of x.

    Reply
  91. Simon Deacon says

    August 11, 2013 at 10:27 am

    Hi Glen – when I posted this it took out the new lines and doesn’t look as neat. I hope you can follow. If not please send your email address through mathswrap and I’ll send a reply.

    Reply
  92. Simon Deacon says

    August 11, 2013 at 10:24 am

    Hi Glen – you’ve got two equations:

    C + Z = 25
    3.2C + 1.4Z = 62

    Multiply first by 3.2 (and leave second) so:
    3.2C + 3.2Z = 80
    3.2C + 1.4Z = 62

    Take second from first, so:
    1.8Z = 18

    Therefore Z = 10

    Then put back into
    C+ Z = 25
    C + 10 = 25

    So C = 15

    The alloy has 15kg of copper and 10kg of zinc.

    I hope this helps and thanks for the question.

    All best

    S

    Reply
  93. Simon Deacon says

    August 11, 2013 at 10:23 am

    Hi Glen – you’ve got two equations:

    C + Z = 25
    3.2C + 1.4Z = 62

    Multiply first by 3.2 (and leave second) so:
    3.2C + 3.2Z = 80
    3.2C + 1.4Z = 62

    Take second from first, so:
    1.8Z = 18

    Therefore Z = 10

    Then put back into
    C+ Z = 25
    C + 10 = 25

    So C = 15

    The alloy has 15kg of copper and 10kg of zinc.

    I hope this helps and thanks for the question.

    All best

    S

    Reply
  94. Glen Rossow says

    August 11, 2013 at 3:26 am

    The materials to make 25kg of an alloy of copper and zinc cost $62. If the copper costs $3.20/kg and the zinc costs $1.40/kg, find the composition of the alloy.

    How would i do that problem?

    Reply
  95. Kim Khus says

    August 8, 2013 at 3:58 pm

    it helps me thanks

    Reply
  96. Simon Deacon says

    July 31, 2013 at 2:43 pm

    Hi – generally yes, although you might need to change the equations a little. Thanks for the comment 🙂

    Reply
  97. zareensaba says

    July 31, 2013 at 12:20 am

    So Mr Simon .if we have two turms negative equation we add after we multiply .and with positive equations we subtract after the multiplication.right?!

    Reply
  98. Devansh Sangoi says

    July 24, 2013 at 3:00 pm

    oregata

    Reply
  99. Simon Deacon says

    July 10, 2013 at 9:15 am

    Hi Ayisha – no, numbers change, although most examples tend to use ‘easier’ numbers. I’ll post a video with some harder questions and let you know when done. All best S 🙂

    Reply
  100. ayisha asuni says

    July 9, 2013 at 11:31 pm

    anytime you have an equation do u always have to use the numbers 3 and 4

    Reply
  101. Simon Deacon says

    June 14, 2013 at 12:31 am

    Hi – yep – went out and bought a better camera soon afterwards! Thanks for the comment and hope the vid was helpful 🙂

    Reply
  102. Ciara Mcsorley says

    June 14, 2013 at 12:23 am

    the only downside is that it isn’t in focus!!

    Reply
  103. Simon Deacon says

    June 2, 2013 at 4:11 pm

    Hi – this would be around B grade. If there was a ‘word’ problem that you needed to create the two equations, it would be an A / B.

    Reply
  104. Georgia Pennant says

    June 2, 2013 at 2:54 pm

    What grade is this?

    Reply
  105. KellyReveiw says

    April 12, 2013 at 1:51 am

    thanks !!! really did help 🙂

    Reply
  106. Simon Deacon says

    March 17, 2013 at 8:20 pm

    Hi … hmm. There could be a way of getting a -7.5 if the width was also negative in your calculation. For most questions they would expect you to then convert to both positive numbers – and say the pool is 7.5m width. My email is on mathswrap – if you send me a copy (photo is fine) of your working – I’ll mark and email back. Hope this helps. S

    Reply
  107. Nada Abdalla says

    March 17, 2013 at 7:57 pm

    would it still be correct if we wrote that the width is equal to -7.5?

    Reply
  108. David Taylor says

    February 16, 2013 at 9:08 am

    The step by step approach is just what students need and the inclusion of real life examples (why we learn this in the first place) is a great bonus.

    Reply
  109. Simon Deacon says

    February 15, 2013 at 2:20 pm

    🙂 thanks

    Reply
  110. lordmoonaz69 says

    February 13, 2013 at 11:21 pm

    Cool, thanks for your videos 🙂 you make it so easy to understand

    Reply
  111. Simon Deacon says

    February 12, 2013 at 11:13 pm

    Hi I’m a maths tutor and also run three first class learning centres.

    Reply
  112. lordmoonaz69 says

    February 12, 2013 at 4:15 pm

    Are you a maths teacher?

    Reply
  113. Simon Deacon says

    February 12, 2013 at 4:14 pm

    Simultaneous word problems coming soon!

    Reply

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