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8th grade (Eureka Math/EngageNY)

Unit 4: Lesson 4

Topic D: Systems of linear equations and their solutions

Systems of equations with substitution: 9x+3y=15 & y-x=5

Learn to solve the system of equations 9x + 3y = 15 and y - x = 5 using substitution. Created by Sal Khan and Monterey Institute for Technology and Education.

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  • blobby green style avatar for user brian10velasquez1
    i am told to solve each system using substitution .
    how would i do this problem ?

    m=2n
    m+4n=30
    (3 votes)
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    • male robot donald style avatar for user Jeremy
      brian,

      Your first equation, m=2n, is already solved for m in terms of n, so plug that in your second equation like this:

      m=2n, so
      m+4n=30 is the same thing as (2n)+4n=30

      Therefore 6n=30, so n= 5. And if n=5 and m=2n, m= 2(5) =10.

      Rechecking:
      m+4n=30. If m=10 and n=5
      (10)+4(5)=30
      10+20=30
      30=30. The answer m=10 and n=5 checks good.

      This method is called solving by substitution, because you first solve for one variable in terms of the other using 1 equation, and then you substitute that solution in the other equation to solve for the second variable explicitly. Then you go back and solve for the first variable explicitly.

      Does that help?
      (11 votes)
  • blobby green style avatar for user Colleen McMonagle
    I have watched almost all the videos for substitution and I still do not understand. For example, for the system of equations x=2y+7, and x=y+4, I got x=-6 and y=-3, but I'm not sure I'm right.
    (2 votes)
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  • piceratops seed style avatar for user Beau Smith
    no one's asked a question in 5+1 years.
    Why is that?
    (3 votes)
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  • blobby green style avatar for user Riley Williams
    how do i solve
    y=x-6
    y=2x-8
    via substitution
    (2 votes)
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    • spunky sam red style avatar for user Marvin Cohen
      Since y = x-6 and y = 2x-8, set x-6 equal to 2x-8.
      x-6 = 2x-8
      -6 = x-8 Subtract x from both sides.
      x = 2 Add 8 to both sides.

      Plug in x for 2 into the first equation y = x-6: y = 2-6 = -4
      Verify that the second equation y = 2x-8 holds true: y = 2(2)-8 = -4, -4 = -4!
      (2 votes)
  • blobby green style avatar for user chiang.ian28
    How do you solve this equation -3x-8y= 20 -5x + y=19
    (2 votes)
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  • blobby green style avatar for user Annie Torres
    How do I solve -8x + y= -3
    3x + 2y =13
    I can't seem to find something similar.
    (2 votes)
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    • piceratops ultimate style avatar for user Justin150
      This problem can be solved using substitution. It takes a few more steps than simpler ones, but here's how you do it:
      The first equations seems simpler, since the y has a coefficient (number before the variable) of 1. If we add -8x to both sides of the equation, we get y = 8x - 3. Now we can substitute this in for y in the second equation to get: 3x + 2(8x-3) = 13, and then we just solve for x. If you did the steps right, you should get x= 1. Then, by putting in 1 for x in any of the two equations, you should get y=5.
      Let's check our work. -8(1)+(5) = -8+5 = -3, and 3(1)+2(5) = 3 + 10 = 13. This is right, so our solution set is x=1, y=5, or (1,5).
      (2 votes)
  • blobby green style avatar for user astallings
    Having trouble solving this
    -4x+y=6
    -5x-y=21
    (2 votes)
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  • blobby blue style avatar for user ✨ Sofia Utama 💯
    Is 0/12 possible? What about 12/0? What's the difference?
    (I am referencing in the video.)

    And if two equations have the same y-intercept (b), does that make the system of equations' point to be that y-intercept point?
    (2 votes)
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  • blobby green style avatar for user I-Guy
    i have this question that is bugging me SO BAD can you help?

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

    Thank You
    (2 votes)
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  • blobby green style avatar for user Becky Ali
    solve using substitution method 6x +3y=12, 9x + 6y=15
    (2 votes)
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    • mr pants teal style avatar for user Anu
      6x+3y=12 ====> 3y=12-6x ===> y=4-2x
      Substitute the value of y in the second equation
      9x+6y=15 (as all the coefficients are multiples of 3, we can divide the whole equation (both the sides) by 3
      3x+2y=5 ===> 3x+2(4-2x) = 5
      3x+8-4x=5 ====> -x = -3 ====> x=3
      Substitute the value of x in the first equation
      y=4-2x =====> 4-2(3) = -2

      Sol : x=3 ; y=-2
      (1 vote)

Video transcript

We're asked to solve and graph this system of equations here. And just as a bit of a review, solving a system of equations really just means figuring out the x and y value that will satisfy both of these equations. And one way to do it is to use one of the equations to solve for either the x or the y, and then substitute for that value in the other one. That makes sure that you're making use of both constraints. So let's start with this bottom equation right here. So we have y minus x is equal to 5. It's pretty straightforward to solve for y here. We just have to add x to both sides of this equation. So add x. And so the left-hand side, these x's cancel out, the negative x and the positive x, and we're left with y is equal to 5 plus x. Now, the whole point of me doing that, is now any time we see a y in the other equation, we can replace it with a 5 plus x. So the other equation was-- let me do it in this orange color-- 9x plus 3y is equal to 15. This second equation told us, if we just rearranged it, that y is equal to 5 plus x, so we can replace y in the second equation with 5 plus x. That makes sure we're making use of both constraints. So let's do that. We're going to replace y with 5 plus x. So this 9x plus 3y equals 15 becomes 9x plus 3 times y. The second equation says y is 5 plus x. So we're going to put 5 plus x there instead of a y. 3 times 5 plus x is equal to 15. And now we can just solve for x. We get 9x plus 3 times 5 is 15 plus 3 times x is 3x is equal to 15. So we can add the 9x and the 3x, so we get 12x, plus 15, is equal to 15. Now we can subtract 15 from both sides, just so you get only x terms on the left-hand side. These guys cancel each other out, and you're left with 12x is equal to 0. Now you divide both sides by 12, and you get x is equal to 0/12, or x is equal to 0. So let me scroll down a little bit. So x is equal to 0. Now if x equals 0, what is y? Well, we could substitute into either one of these equations up here. If we substitute x equals 0 in this first equation, you get 9 times 0 plus 3y is equal to 15. Or that's just a 0, so you get 3y is equal to 15. Divide both sides by 3, you get y is equal to 15/3 or 5. y is equal to 5. And we can test that that also satisfies this equation. y, 5, minus 0 is also equal to 5. So the value x is equal to-- I'll do this in green. x is equal to 0, y is equal to 5 satisfies both of these equations. So we've done the first part. Let's do the second part, where we're asked to graph it. The second equation is pretty straightforward to graph. We actually ended up putting it in mx plus b form right there. And actually, let me rewrite it. Let me just switch the x and the 5, so it really is in that mx plus b form. So y is equal to x plus 5. So its y-intercept is 5-- 1, 2, 3, 4, 5-- and its slope is 1, right? There's a 1 implicitly being multiplied, or the x is being multiplied by 1. So it looks like Let me see how well I can draw it. The line will look like that. It has a slope of 1. You move back 1, you go down 1. You move forward 1, you go up 1. That's a pretty good job. So that right here is this equation. Now let's graph that top equation. And we just have to put it in mx plus b form, or slope-intercept form. And I'll do that in green. So we have 9x plus 3y is equal to 15. One simplification we can do right from the get-go is every number here is divisible by 3, so let's just divide everything by 3 to make things simpler. So we get 3x plus y is equal to 5. Now we can subtract 3x from both sides. We are left with y is equal to negative 3x plus 5. So that's what this first equation gets turned into, if you put it in slope-intercept form. y is equal to negative 3x plus 5. So if you were to graph it, the y-intercept is 5. 0, 5. And then its slope is negative 3. So you move 1 in the x-direction, you move down 3 in the y-direction. Move 2, you would move down 6. 2, 4, 6. Move 2, you go 2, 4, 6. So this line is going to look something like this. It's going to look something like that right there. As you can see, the solution to this system is the point of intersection of these two lines. It's the combination of x and y that satisfy both of these. Remember, this pink line, or this red line, is all of the x's and y's that satisfy this equation: y minus x is equal to 5. This green line is all of the x's and y's, or all the combinations of them, that satisfy this first equation. Now, the one x and y combination that satisfies both is their point of intersection. And we figured it out algebraically using substitution. That happens at x is equal to 0, y is equal to 5. x is equal to 0-- this is the x-axis-- y is equal to 5 right there.