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Worked example: evaluating expressions using structure (more examples)

A few more examples where Sal evaluates expressions with multiple unknown variables. Created by Sal Khan.

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  • blobby green style avatar for user karandeep kullar
    I think there is solution to the equation being talked about at . 3a+5b=2. 15a+15b=?
    I am not too sure I double checked and it seems correct.
    = 3(5a+5b)
    = 3(2a+3b+5b)
    = 3(2a+2)
    = 6a+6
    a = -6/6
    a = -1


    15(-1)+15(1) = ?
    -15+15 = 0
    (6 votes)
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    • leaf grey style avatar for user jellback
      You did some really clever factorisation but there are some pretty clever errors as well. Lets review the first part where you got a = -1.
      At the start you write = 3(5a+5a), but what does it equal? The biggest error comes from this statement.

      Lets replace the question mark in the equation with c, this seems proper, so:
      15a + 15b = c, this is how the first line should look.
      3(5a + 5b) = c
      3(2a + 3a + 5b) = c, you did make a typo here but this step was cool.
      3(2a + 2) = c
      6a + 6 = c
      In the last 2 steps of your calculation of a you put c as 0, but we can't make that assumption out of the blue, c can be anything. In the last equation I wrote we see that if we know a we can calculate c or vice versa. But to know a we need b.

      At this point we know that to solve this problem we need more information!
      (6 votes)
  • male robot hal style avatar for user Gerry Tipple
    The 'insoluble' 3a + 5b = 2 question. Why not divide both sides by the common denominator of 3 and 5, i.e. 15, giving after simplification, a +b = 2/15
    (4 votes)
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  • blobby green style avatar for user Stacey Broomell
    How do I evaluate an equation like y+7/x-5=2/3?
    (3 votes)
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  • leaf red style avatar for user Hann

    It led to infinite solution. Hence, not enough information, am I correct?
    (1 vote)
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  • leaf green style avatar for user Earl Tilluck
    I think I solved the last one, but I might be wrong:
    For the last question I was able to do the following

    3a + 5b = 2
    15a + 15b = ?

    consider the following:

    x(3a+5b) = 15a +15b
    (I'm asking what number times the first equation is equal to the second equation)

    x(3a + 5b) = 15a +15b
    x = 15a +15b / 3a + 5b
    x = 5a +3b / 2


    (3a + 5b) x ( (5a + 3b) / 2 ) = 15a + 15b

    intuition tells me that:
    3a+5b = 5a+3b
    because of the rules of multiplication and addition, so.....
    3a+5b = 2
    5a+3b = 2
    if we substitute these values we get:

    2 x (2/2) = 15a + 15b
    15a + 15b = 2

    so the answer is "2"
    Is this true or did I do something wrong?
    (0 votes)
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    • leaf blue style avatar for user Matthew Hochberg
      The problem with your assessment is that 3a+5b = 5a+3b is not really true. You are probably thinking of the Commutative Property which says you can move around numbers when adding them. So 3a+5b = 5b+3a, the variables stay with their coefficients, you cannot move the variables around as if they are separate from the numbers they are being multiplied by.
      The reason why we can't solve for 3a + 5b = 2 and 15a + 15b = ? is because the ? mark should be another variable as x. The equations would become 3a + 5b = 2 and 15a + 15b = x, and there is a little rule in algebra that you need at least the same number of equations as the number of variables we are solving for. Hence two equations are not enough for three variables.
      Final note, your writing of
      x(3a + 5b) = 15a +15b
      x = 15a +15b / 3a + 5b
      x = 5a +3b / 2
      has a little error where changing 15a +15b into 5a +3b doesn't work because you can't divide the coefficients of each variable directly (15 divided by 3 and then by 5). In fact the equation should have be written as
      x = (15a +15b) / (3a + 5b)
      x = (15a +15b) / (2)
      x = (15/2)a + (15/2)b
      x = (15/2)(a + b)
      and we are stuck in the same situation as in the video with x = (15)(a + b).
      Hope this was helpful.
      (4 votes)
  • piceratops ultimate style avatar for user Dr. Blade
    Whilst doing this exercise, I got the question "If 8a+2b+6c= -6, what is 16a+4b+12c?." Can I factor out "8a+2b+6c= -6" by doing 8+2+6(a+b+c)= -6, eventually getting -2 2/3 for (a+b+c)?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      I don't think your method will work. The way I do these is I compare: 8a+2b+6c to 16a+4b+12c. What can you multiply with 8a+2b+6c to get to 16a+4b+12c? Well, 8a can become 16a if we multiply by 2. Does that work for all the terms? Let's see. 2b can become 4b also by multiplying by 2. And, 6c becomes 12c by multiplying by 2.

      We now know: 2 (8a+2b+6c) = 16a+4b+12c. Thus, 2(-6) must also = 16a+4b+12c. Answer is: -12.
      (2 votes)
  • ohnoes default style avatar for user Humza Khan
    At , why did sal cross out the expression divided by 3?
    (1 vote)
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    • stelly blue style avatar for user Kim Seidel
      He's dividing both sides of the equation by 3. 3 divided by 3 = 1. And, if you multiply anything by 1, it doesn't change the other value (5*1 = 5; x*1 = x; etc).

      You can also look at it as he is reducing the fraction. This process of cancelling common factors is done all the time when reducing fractions.
      (2 votes)
  • leaf blue style avatar for user vaishnavi.tetali
    Um, I was just wondering couldn't you do the first problem:3x+3y+3z=1 and 12x+12y+12z=?
    on a different way?
    I solved the problem by dividing equation 2 by equation 1 so i got um... 4x+4y+4z=4
    Or you could of just 4(3x+3y+3z=1)===== 12x+12y+12z=4!! Therfore the 4 would replace the question mark!!
    I got the same answer as Sal but i did it in a different way so i was just wondering if this could work?Does it work all the time? Sorry if this confused you!
    (1 vote)
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  • spunky sam blue style avatar for user mtchuenkam
    in the first problem I didn't understand why sal divided 3 from both sides of 3(x+y+z) equals 1. can someone please explain?
    (1 vote)
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    • hopper jumping style avatar for user Michael Laman
      Sal sees a common structure to both equations. Getting them to something in common let him use information from one equation to solve the other. The common structure in the two equations is "a + b + c". If he can both equations to that term with whatever else remains getting it there, he can use one equation to solve the other.

      In this example he takes the approach of getting to a common "a + b + c" in both equations. Again, then he can use one of them to substitute in the other.

      3x + 3y + 3z = 1 [ Original equation. Want to get it to "x + y + z" so it can be substituted in the second equation ]
      3(x + y + z) = 1 [ Factored out a common 3 from each of the first here terms. ]
      (x + y + z = 1/3 [ Divide both sides by 3 to get x + y +z by itself. ]

      Now he can replace "x + y + z" by 1/3 in the second equation to end with 4.

      I hope this helps.
      (2 votes)
  • spunky sam blue style avatar for user Samir Gunic
    How do you do the addition -8 + 30? I like to think of the difference in absolute value, and then I look at what number is smaller and what number is bigger and try to think of it as number line and figure out if I will go to the right of zero or to the left of zero.

    I recently heard Sal say in a video that he likes to rewrite these, I think he distributed out the negative sign somehow and he had two numbers in the parentheses. But I got wrong answer because of this.

    -8+30 is not - (30-8) I know that now. The only way I can think of is -(-30+8).
    (1 vote)
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Video transcript

Let's do a few more examples where we're evaluating expressions with unknown variables. So this first one we're told 3x plus 3y plus 3z is equal to 1, and then we're asked what's 12x plus 12y plus 12z equal to? And I'll give you a few moments to think about that. Well let's rewrite this second expression by factoring out the 12, so we get 12 times x plus y plus z. That's this second expression here, and you can verify that by distributing the 12. You'll get exactly this right up here. Now, what is 12 times x plus y plus z? Well, we don't know yet exactly what x plus y plus z is equal to, but this first equation might help us. This first equation, we can rewrite this left-hand side by factoring out the 3, so we could rewrite this as 3 times x plus y plus z is equal to 1. All I did is I factored the 3 out on the left-hand side. And then if I want to solve for x plus y plus z, I just divide both sides of this equation by 3, and I'm left with x plus y plus z is equal to 1/3, and so here, instead of x plus y plus z, I can write 1/3. So this whole thing simplified to 12 times 1/3. 12 times 1/3 is the same thing as 12 divided by 3, which is equal to 4. Let's try one more. So here we are told that 3a plus 5b is equal to 2, and then we're asked what's 15a plus 15b going to be equal to? So we might-- let's see. I'll give you a few moments to try to tackle this on your own. Let's see how we might do it. We could approach it the way we've approached the last few problems, trying to rewrite the second expression. We could rewrite it as 15 times a plus b, and so we just have to figure out what a plus b is, and we'll be able to evaluate this expression. And so, it's tempting to look up here, and say maybe we can solve for a plus b somehow, but we really can't. If we divide-- if we try to factor out a 3, we'll get 3 times a plus 5/3b, so this doesn't really simplify things in terms of a plus b. If we try to factor out a 5, we'd get 5 times 3/5a plus b is equal to 2, but neither of these gets us in a form where we can then solve for a plus b. So in this situation, we actually do not have enough information to solve this problem. So it's a little bit of a trick. Not enough info to solve. Anyway, hopefully you enjoyed that.