How to solve simultaneous linear equations … using algebra

February 15, 2013128 Comments

How to solve simultaneous linear equations.. using algebra…

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 B) and involves dealing with a negative term.
  • The third video … shows more of a real application..

Click here for simultaneous linear equations quick test

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: Simultaneous pencils

  • If you sell at a high price they’ll be less demand
    Simultaneous pencils
  • 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 

How to solve simultaneous equation word problems 


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Comments (128)

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  1. Renea Chua says:

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

  2. MagicMinter HD says:

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


  3. InsaneGamer says:

    where have you been all this time??

  4. SuperSonicSpeedGuy says:

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

  5. Hannah Dale says:

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

  6. sandev randhawa says:

    great video to watch

  7. Ajit King says:

    did you know science

  8. Ajit King says:

    you are a very good teacher dude.

  9. CoolCat4196 says:

    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

  10. kurtis sankey says:

    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?

  11. Ahmed El-Aide says:

    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

  12. J. Moore says:


  13. unknown profile says:

    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 :)

  14. Lee Dodd says:

    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.

  15. The Astute Atheist. says:

    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:

    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.

    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.

  16. ABDUL JAWAD Ghalib says:

    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?

  17. On The Road Media says:

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

  18. willoxens says:

    Could you help me with a question?
    It asks to consider the equations:
    where k is a real constant for which I have to find the unique solution.
    Thanks in advance!

  19. Ambra Onyss says:

    where did x=4 come from?

  20. yassin ramadan says:

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

  21. hi ih says:

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

  22. CoD Shotz says:

    Thank you this was really helpful.

  23. stephen barker says:

    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

  24. Shoot Fresh Mine says:

    hey can u help me answer these questions





  25. Shoot Fresh Mine says:

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


  26. Dafydd Morris says:

    How would I do

  27. Dafydd Morris says:

    Praise be Simon deacon 😀

  28. Shoot Fresh Mine says:

    How would u work out


  29. Cethical says:

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

  30. ABWEndon says:

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

  31. vani na says:

    thank you so much

  32. Phase Gaming says:

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

  33. stephen barker says:

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

  34. Mathew Stenton says:

    Wow Simon your my best friend and I love you thanks

  35. OpiumMessiah says:

    Thanks, man. :)

  36. Phoenix Feathers says:

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

  37. Charlie Michele says:

    Thank you so much! I finally get it!

  38. Daniel Gadd says:

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

    Help me here someone please

  39. Daniel Kibby says:

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

  40. Walter Payton says:

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

  41. MegaSaya64 says:


  42. Hayden Nicholson says:

    Thanks +Simon Deacon helped alot!

  43. bubbles xo says:

    very helpful cheers

  44. Marcin Szostok says:


  45. thejchan16 says:

    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!

  46. Michelle Wanha says:

    Thank you sir!

  47. Mateen Malik says:

    Thanks makes a lot more sense.

  48. Jose Dominguez says:

    Fix the audio

  49. ibraheem nawaz says:

    Awesome Best tutor you are better than my maths teacher.. :)

  50. Tendaishe Mbele says:

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

  51. stephen barker says:

    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.

  52. Matthew Sydoruk says:


  53. stephen barker says:

    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

  54. yukicross6767 says:

    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

  55. Peter Backstrand says:

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

  56. stephen barker says:

    how do you solve simultanious equations graphically

  57. Jibrael Malik says:

    Thanks helped me out!

  58. stephen barker says:

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

  59. HilalFener says:

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

  60. stephen barker says:

    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

  61. stephen barker says:

    youll have do some with minus terms

  62. stephen barker says:

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

  63. Josh Shil says:

    Hi Simon Deacon I am looking for questions that give the answer of both x’s

  64. stephen barker says:

    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

  65. Cameron Richmond says:

    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

  66. davidymcmb says:

    Thanks for the help man!

  67. stephen barker says:

    ive nailed equations with minus numbers already for next year now

  68. BAT TT says:

    This is very helpful thank you

  69. Justin Beary says:

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

  70. Annie Lim says:

    it is really blur

  71. mussa sherif says:


  72. killa killa says:

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

  73. stephen barker says:

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

  74. mohamed abdelhalim says:

    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

  75. SkizaKiza says:

    thankyou for the help!

  76. Simon Deacon says:

    Please like and leave a comment!

    Visit for real maths, tips and techniques.

  77. Satellite Dave says:

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

  78. Finlay McKee says:

    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!

  79. Anita Cheung says:


  80. Simon Deacon says:

    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

  81. stephen barker says:

    a graph how would i do that

  82. Simon Deacon says:

    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!

  83. stephen barker says:

    whats the easyest way to solve these

  84. Simon Deacon says:

    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

  85. mymindydiditong says:


  86. mymindydiditong says:

    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?

  87. Simon Deacon says:

    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

  88. mustanser hussain says:


  89. mustanser hussain says:

    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.

  90. Simon Deacon says:

    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.

  91. Simon Deacon says:

    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


  92. Simon Deacon says:

    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


  93. Glen Rossow says:

    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?

  94. Kim Khus says:

    it helps me thanks

  95. Simon Deacon says:

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

  96. zareensaba says:

    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?!

  97. Devansh Sangoi says:


  98. Simon Deacon says:

    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 :-)

  99. ayisha asuni says:

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

  100. Simon Deacon says:

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

  101. Ciara Mcsorley says:

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

  102. Simon Deacon says:

    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.

  103. Georgia Pennant says:

    What grade is this?

  104. KellyReveiw says:

    thanks !!! really did help :)

  105. Simon Deacon says:

    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

  106. Nada Abdalla says:

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

  107. David Taylor says:

    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.

  108. Simon Deacon says:

    :-) thanks

  109. lordmoonaz69 says:

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

  110. Simon Deacon says:

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

  111. lordmoonaz69 says:

    Are you a maths teacher?

  112. Simon Deacon says:

    Simultaneous word problems coming soon!

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