If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

Main content

Substitution method review (systems of equations)

The substitution method is a technique for solving a system of equations. This article reviews the technique with multiple examples and some practice problems for you to try on your own.

What is the substitution method?

The substitution method is a technique for solving systems of linear equations. Let's walk through a couple of examples.

Example 1

We're asked to solve this system of equations:
3x+y=3x=y+3
The second equation is solved for x, so we can substitute the expression y+3 in for x in the first equation:
3x+y=33(y+3)+y=33y+9+y=32y=12y=6
Plugging this value back into one of our original equations, say x=y+3, we solve for the other variable:
x=y+3x=(6)+3x=3
The solution to the system of equations is x=3, y=6.
We can check our work by plugging these numbers back into the original equations. Let's try 3x+y=3.
3x+y=33(3)+6=?39+6=?33=3
Yes, our solution checks out.

Example 2

We're asked to solve this system of equations:
7x+10y=362x+y=9
In order to use the substitution method, we'll need to solve for either x or y in one of the equations. Let's solve for y in the second equation:
2x+y=9y=2x+9
Now we can substitute the expression 2x+9 in for y in the first equation of our system:
7x+10y=367x+10(2x+9)=367x+20x+90=3627x+90=363x+10=43x=6x=2
Plugging this value back into one of our original equations, say y=2x+9, we solve for the other variable:
y=2x+9y=2(2)+9y=4+9y=5
The solution to the system of equations is x=2, y=5.
Want to learn more about the substitution method? Check out this video.

Practice

Problem 1
Solve the following system of equations.
5x+4y=3x=2y15
x=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi
y=
  • Your answer should be
  • an integer, like 6
  • a simplified proper fraction, like 3/5
  • a simplified improper fraction, like 7/4
  • a mixed number, like 1 3/4
  • an exact decimal, like 0.75
  • a multiple of pi, like 12 pi or 2/3 pi

Want more practice? Check out this exercise.

Want to join the conversation?

  • old spice man blue style avatar for user Matt
    I am curious if there are times when either the elimination method or the substitution method would be more appropriate, and or if there would be times when only one way or the other would work. Thank you for the advice in advance!
    (54 votes)
    Default Khan Academy avatar avatar for user
    • orange juice squid orange style avatar for user Matthew Johnson
      Yes, both equations will always work, but yes, at times it is more logical to use one over the other. for instance:
      x-4y=6
      x+4y=12

      we see here that elimination would best fit:
      x+x+4y-4y=6+12


      or:
      x=2y-3
      7x+3y-81=23

      substitute (2y-3) for x.

      Also, once you have a single placeholder, put it ito quadratic form (ax^2+bx+c=0)
      and use the quadratic formula:
      x=(-b±sqrt(b^2-4ac))/2a
      (38 votes)
  • blobby green style avatar for user David
    This is pretty challenging not gonna lie
    (56 votes)
    Default Khan Academy avatar avatar for user
  • blobby green style avatar for user Nancy Crisp
    how do I solve y=2x-1 and y=3x+2 using the substitution method
    (8 votes)
    Default Khan Academy avatar avatar for user
  • stelly blue style avatar for user kenaniah
    This is so hard I can't.
    (8 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      This is the review. Start at the beginning of the lesson. Go thru each video step by step and make sure you understand before moving to the next one. Ask specific questions when there is some part of the video that you don't understand.

      As you do the practice problems, if you get one wrong, use the hints to learn from your mistakes.

      Then, come back and try the review again.
      (13 votes)
  • leafers seed style avatar for user Theo  Lesko
    are there any easy tips or tricks i can use to remember this?
    (10 votes)
    Default Khan Academy avatar avatar for user
  • leafers ultimate style avatar for user joshh
    Practice Solutions:

    First system of equations:

    1st equation: -5x + 4y = 3
    2nd equation: x = 2y - 15

    Substitute for x:
    -5(2y - 15) + 4y = 3

    Solve for y:
    -10y + 75 + 4y = 3
    -6y = -72
    y = 12

    Solve for x by substituting in one of the equations:
    x = 2y - 15
    x = 2(12) - 15
    x = 9

    Check your solution by substituting for x and y in the first equation:

    -5x + 4y = 3
    -5(9) + 4(12) = 3
    -45 + 48 = 3
    3 = 3 √


    Second system of equations:

    5x - 7y = 58

    y = -x + 2

    Subsitute for y:
    5x - 7(-x + 2) = 58

    Solve for x:
    5x + 7x - 14 = 58
    12x - 14 = 58
    12x = 58 + 14
    x = 72/12
    x = 6

    Solve for y by substituting in one of the equations:
    y = -x + 2
    y -(6) + 2
    y = -4


    Check your solution by substituting for x and y in the first equation:

    5x - 7y = 58
    5(6) - 7(-4) = 58
    30 + 28 = 58
    58 = 58 √


    I hope this could help someone. It seems like a lot, but once you understand the steps and get the hang of it, it will become pretty quick!
    (10 votes)
    Default Khan Academy avatar avatar for user
  • starky sapling style avatar for user CleverDesire
    If i may ask,could anyone help me in this equation:

    3y+2x=7
    y-3x=6

    Please answer the question in with explaination.thanks🙏🏽🥺😊
    (4 votes)
    Default Khan Academy avatar avatar for user
    • stelly blue style avatar for user Kim Seidel
      I'll get you started.
      Take the 2nd equation and add 3x to both sides to isolate "y".
      y = 3x+3
      Now, substitute this value of "y" into the first equation.
      3(3x+6) + 2x = 7
      You can now solve for "x".
      Once you have the value of "x", use it to calculate the value of "y".

      Hope this helps. Comment back if you have questions.
      (8 votes)
  • hopper cool style avatar for user Jaiden Galecki
    who is up for quiz one
    (8 votes)
    Default Khan Academy avatar avatar for user
  • mr pants teal style avatar for user Emmery
    In example 2, when it says we have to solve for x or y, how do I get -2x+y=9 into slope intercept form? And also, how did we get rid of the negative once it was in slope intercept form?
    Thanks
    (5 votes)
    Default Khan Academy avatar avatar for user
  • stelly green style avatar for user LeeAnn  Morales
    Lol this is actually kind of simple, though it takes a different mindset. I like to think of the first equation as the problem I'm trying to solve and the second as a hint. See the second one says what x equals even if it doesn't necessarily give you the answer. Because it doesn't give the answer for y, we have to replace the x with the second equation to be able to solve for it. After solving for y, you would plug y into one of the two original equations and solve for x the way you did for y. After you solve for x, you would have the two answers you needed, x and y :) (I hope this was helpful, sorry if it wasn't :[ ...)
    (7 votes)
    Default Khan Academy avatar avatar for user