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Systems of equations with substitution: y=4x-17.5 & y+2x=6.5

Learn to solve the system of equations y = 4x - 17.5 and y + 2x = 6.5 using substitution. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user Dolotinab
    Are you supposed to use the substitution method whenever you see the systems of equations?
    (5 votes)
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    • orange juice squid orange style avatar for user Jared
      Systems of equations can be solved by graphing, elimination, or substitution. Which one you use comes down to the systems of equations you're solving for, and how much work you want to do. Some systems solve easier using elimination, while others solve quickly using substitution.
      (6 votes)
  • spunky sam blue style avatar for user Brianna J.
    What if the problem is

    -5x + y= -2

    -3 + 6y= -12

    How would I figure out either the x value first or the y value first ?
    (4 votes)
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    • blobby green style avatar for user skeit079
      You were already given the Y value, look:

      -5x + y = -2 -3 + 6y = -12
      y = -2 + 5x 6y = -9
      y = -9/6
      y = -3/2 <--- this is your Y value

      Y = -9/3 (which is the same as -3/2), all you have to do to solve for X is substitute your Y value into the equation: -5x + 7 = -2

      Here it goes:

      -5x - 2/3 = -2
      -5x = -2 +(2/3)
      -5x = -4/3
      x = (-4/3
      (4 votes)
  • blobby green style avatar for user P B
    how would you solve this problem
    x+y+7z=21
    x+y+9z=29
    x+y+4z=39
    (4 votes)
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    • male robot hal style avatar for user Sid
      If you subtract the first one from the second one, you find that z = 4.
      If you subtract the third one from the second one, you find that z = -2.
      So, those equations don't have a solution.
      (2 votes)
  • spunky sam blue style avatar for user Brianna J.
    what if you have a problem like this:

    -3x - 4y = 2

    3x + 3y = -3

    ?
    (2 votes)
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  • blobby green style avatar for user justinheard
    Solve by substitution.
    System A and System B
    3x-5y=13
    2x+y=4
    (2 votes)
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    • old spice man green style avatar for user newbarker
      In the second equation, isolate y on the left by subtracting 2x from both sides. Now plug the new right hand side of equation two (i.e. 4 - 2x) for y in the top equation. The top equation is now in just one variable: x. Solve this for x. Plug this value for x into either equation and solve for y.

      Please comment with your answer or any problems you run into.
      (2 votes)
  • blobby green style avatar for user gabrielle barton
    How do you solve by substitiution??
    3x-2y=7
    x+2y=5
    (2 votes)
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  • blobby green style avatar for user jennyfer mendoza
    I don't understand how to make one that looks like this:


    -5x+3y=1
    4x+y=23
    (1 vote)
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  • piceratops seedling style avatar for user Jessica Torres
    what if you were to have -7x+8y=-5/ -2x+6y=6?
    (1 vote)
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    • piceratops ultimate style avatar for user Just Keith
      -7x+8y=-5 AND -2x+6y=6 Solving by substitution:
      Step 1: solve one of the equations for one of the variables. Pick either equation and either variable, just whichever is convenient.
      -2x+6y=6
      Divide both sides by -2 to get: x - 3y = - 3
      Add 3y to both sides to get: x = 3y - 3
      Use this to substitute for x into the other equation:
      -7x+8y=-5
      -7(3y - 3) + 8y = -5
      -21y + 21 + 8y = -5
      Add -21 to both sides to get: -21y + 8y = -26
      Combine like terms to get: -13y = -26
      Divide both sides by -13 to get: y = 2
      Plug y= 2 into one of the equations and solve for x:
      x = 3y - 3
      x = 3(2) - 3
      x = 3
      (4 votes)
  • blobby green style avatar for user Laeeqa Ahmad
    The problem is 840=(x+y) eq.1 & 840=(y-x)7 need to understand how to solve in substitution method HELP?
    (2 votes)
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  • mr pink red style avatar for user muth0821
    how do i solve this
    2x-3y=-24
    x+6y=18
    (1 vote)
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    • old spice man green style avatar for user t.rybner
      Try watching Sal's videos again before reading on.
      Convert either of the equations into the form x = a*y + b or y = c*x + d
      Substitute the value for x (or y) in the other equation
      Solve this equation
      Plug the result into either of the original equations
      Solve that
      Voila

      So you could start by converting
      x + 6y = 18
      into
      x = -6y + 18

      Plug that into 2x - 3y = -24
      2*(-6y + 18) - 3y = -24
      Multiply into brackets
      -12y + 36 - 3y = -24
      Solve this equation for y
      Plug the result into x = -6y + 18 to get x
      (2 votes)

Video transcript

We have this system of equations, y is equal to 4x minus 17.5, and y plus 2x is equal to 6.5. And we have to solve for x and y. So we're looking for x's and y's that satisfy both of these equations. Now, the easiest way to think about it is we've already solved for y in this top equation. Let me write it again. I'll write it in pink. We have y is equal to 4x minus 17.5. So this first equation is telling us, literally, by this constraint, y should be 4 times x minus 17.5. Now, the second equation says whatever y is, we had 2 times x, and that should be 6.5. Well, the y here also has to meet this constraint up here. It also has to meet the constraint that it has to be 4 times x minus 17.5. So what we can do is, is we can substitute this value for y into this equation. Let me be clear what I'm doing. The second equation here is y plus 2x is equal to 6.5. We know that y has to be equal to this thing right here. y has to be equal to 4x minus 17.5. So let's take 4x minus 17.5, and substitute y with that. So let's put that right there. So if we were to do that, if we were to replace this y with 4x minus 17.5, because that's what the first equation is telling us, then we get 4x minus 17.5, plus 2x is equal to 6.5. And now we have a single linear equation with one unknown. Let's solve for x. So first we have our x terms. We have a 4x, and we have a 2x. We can group them or add them together. 4x plus 2x is 6x. And then we have 6x minus 17.5 is equal to 6.5. Then we can get the 17.5 out of the way by adding it to both sides of the equation. So this is negative 17.5, so let's add positive 17.5 to both sides of this equation. And we are left with the left-hand side is just going to be 6x, because these guys cancel out. 6x is going to be equal to-- and 6.5-- see, 6 plus 17 is 23, and then 0.5 plus 0.5 is 1. So this is going to be 24. And then we can divide both sides of this equation by 6. And you are left with x is equal to 24 over 6, which is the same thing as 4. So we figured out the x value for the x and y pair that satisfy both of these equations. Now we need to figure out the y value. And we can do that by taking this x and putting it back into one of these equations. We can do it in to either one. We should get the same y value. So let's just do this top one up here. So if we assume x is equal to 4, this top equation tells us y is equal to 4 times x, which in this case is 4, minus 17.5. Well, this is equal to 16 minus 17.5, which is equal to negative 1.5. So y is equal to negative 1.5. So the solution to this system is x is equal to 4, y is equal to negative 1.5. And you can even verify that these two, they definitely work for the top one if you put 4 times 4, minus 17.5, you get negative 1.5. But they also work for the second one. And let's do that. In the second one, if you take negative 1.5, plus 2 times x-- plus 2 times 4-- what does that equal? That's negative 1.5 plus 8. Well, negative 1.5 plus 8 is 6.5. So this x and y satisfy both of these equations.