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### Course: Algebra (all content)>Unit 10

Lesson 13: Evaluating expressions with unknown variables

# Worked example: evaluating expressions using structure

Sal solved a few examples where we need to evaluate an expression, but we don't know the individual variable values. Created by Sal Khan.

## Want to join the conversation?

• Hi, I'm in 8th grade and have a cumulative test soon. On my advanced supplement, I came across a fairly hard question (ha, it's probably not very hard for all the smart people on this website :D). I found an answer, but a is in terms of r and b. I will certainly ask my teacher later, but it would be nice and helpful if I could get an explanation here. So: A line L has a slope of -2 and passes through the point (r,-3). A second line, K, is perpendicular to L at (a,b) and passes through the point (6,r). Find a in terms of r. (sorry if this is in the wrong category, but I wasn't entirely sure which one it should be in)
• at you say you can factor out a -9 but why is it a negative? If the original equation is a+b+c = -1 then you would need to factor in a positive 9, right? I'm so confused
• can i just say probably try to rewatch the video and pay attentiion close to what he is saying and watch his steps and if that still doesnt answer your question try to research it
• if the student's still don't understand how can i make it easer for them?
(1 vote)
• Here is what I'd suggest:
Rewatch the videos.
Ask questions (there are people like me who do know this stuff who can help) -- hint, ask very specific questions.
If you still struggle, there are other websites that have educational videos on beginning algebra. There are plenty on YouTube (some are videos of actual teachers teaching a real classroom).
You might see if you can get an Algebra workbook to practice this material with -- Amazon.com and other venues have such workbooks for not very much money.
• where did you get the 5 out of
5a+5b+5C?
• 5 a + 5 b + 5 c
Like Sal explains at around when you rewrite the expression as 5 (a+b+c) its the same thing as 5 a + 5 b + 5 c because of the distributive property.
Hope I helped!
• what if a + 2b + 5c = 7 and 4a + 8b + 20c = d. then what does d = _ ?
• d would be equal to 4a+8b+20c or 4(a+2b+5c) or 28
• So if my question is 2r+3r-4-8=r-8 how am I going to solve that
• How can you tell if the number is a negative or if it is subtracting? For instance, if the problem was:
-9b-9c-7d, how do you know if the 9c and 7d are negatives or being subtracted? In the video, Sal said all the terms were negatives, but how did he know?
(1 vote)
• Sometimes, negative numbers have a "-" sign before it like -1 and subtracting is like - by itself if you see a sentence like.....
-3 - -2 =?
That basically means -3 (-) -2=? so if its like a math test just do a ( ) over the subtracting sign and if you can't tell which is which still look at the number after the negative - 3 which in this case 3 then look at - which is (-) and the last one is -2 so its -3(-)-2= -5!
HOPE IT HELPS ALSO CHECK OUT THIS WEBSITE:

http://www.mathsisfun.com/positive-negative-integers.html
• Hi. At about , Sal shows us that the equation can be solved liked this -
5(7)

So does that mean that a+b+c=7 = a=0, b=0, c=7?
Couldn't a=1, b=1, c=5 or another combination? Thanks.
• You're doing things a little different than Sal.

Sal is solving them with coefficients attached, be they a 7 or a -9. If we know what a, b, and c are, then we can solve it by simply plugging in the numbers. However, if we only know that a + b + c is 7, then we can apply that to all other questions with coefficients in them. In other words, if a, b, and c are unknowns, then we have to solve with the information we have. We are generalizing. In the first half of the video, we see that 5a + 5b + 5c = 35; a, b, and c could be anything as long as they equal 35.

So if they add up, a, b, and c could be anything, really.

Hope that helps!
(1 vote)
• at Sal says he "can factor out and get a -9". Is there a video I missed somewhere here explaining that? I understand factoring, but just not how and why it gets applied in these algebraic expressions. Thanks
• As long as you don't change the underlying value of the expression, you can use arithmetic properties to get equivalent versions of the expression. Sal is using the distributive property to get part of the expressions to just be: a+b+c and another part to be x+y. If he can accomplish that, he can use substitution and actually evaluate the expression.

If you have factored polynomials with 4 terms, you have likely used the distributive property in a similar fashion. Here's a lesson on that topic that might help: https://www.khanacademy.org/math/algebra/polynomial-factorization/factoring-quadratics-2/a/factoring-by-grouping
(1 vote)
• What if we don't know what the variables give us

## Video transcript

I'm now going to do a few classic problems that show up a lot at math competitions and sometimes on standardized tests, and they seem like very daunting and intimidating problems. But hopefully over the course of this video, you'll realize that if you kind of see what they're asking for, it's really not that bad. So let's try this first one right over here. They tell us a plus b plus c is equal to 7. And then they ask us, what is 5a plus 5b plus 5c equal to? And the first reaction's like, well, they just gave me one equation right over here. One equation with three unknowns, how do I solve for a, b, or c? Don't I have to be able to solve for a, b, or c in order to figure out what 5a plus 5b plus 5c is? So I'll let you think about that for a second. So the big idea here is to realize that 5a plus 5b plus 5c, that's just the same thing as 5 times a plus b plus c. If you distribute the 5 here, 5 times a, 5 times b, 5 times c, you get 5a plus 5b plus 5c. Another way you could think of it is we are factoring a 5 out. If you factor a 5 out, you're left with 5 times a plus b plus c. Now, how do we evaluate this? Well, that first equation gave us all the information that we needed. They told us that a plus b plus c is equal to 7. So 5 times a plus b plus c is the same exact thing. I think you see where this is going now. It's 5 times 7, and now this becomes pretty straightforward. That's going to be equal to 35. Let's do one more of these. And this is going to be a little bit more involved, but hopefully you see the same idea. So we're told a plus b plus c is equal to negative 1. And we have two more variables here. We're told that x plus y is equal to 7. Then they ask us this big, hairy thing. What is this equal to? And I'll give you a few seconds to give it a little bit of thought. Well, you could imagine. They've given three variables with one equation up here, another two variables with another equation. We have five unknowns with two equations. There's no way we're going to be able to individually figure out what a, or b, or c, or x, or y is. But maybe we can use what we saw in the last example to solve this. And so what we might want do is rearrange it so that we have the x's and the y's kind of grouped together, and the a's, b'c, and c's kind of grouped together. So let's try that out. So let's focus first on the a's, b's, and c's. So we have negative 9a, negative 9b-- I'll go in order, alphabetical order-- negative 9b, and we have negative 9c, and I think you see what's emerging. And then let's work on the x's and the y's. And then we have negative 7x, minus 7x, and then we have a minus 7y. So all I have done is rearranged this expression here. But this makes it a little bit clearer of what's going on here. These first three terms, I can factor out a negative 9, and I get negative 9 times a plus b plus c. And these second two terms, I can factor out a 7, minus 7 times x plus y. And just to verify, if you wanted to go the other way, multiply this negative 7 times x plus y, you'll get this. Multiply negative 9 times a plus b plus c, distribute it, you'll get this right over here. And so this makes it a little bit clearer. What is a plus b plus c equal to? Well, they tell us right over here. a plus b plus c is equal to negative 1. This whole expression is negative 1, at least in the parentheses. And what is x plus y equal to? Let me do this in a new color. What is x plus y equal to? Well, they tell us right over here. x plus y is equal to 7. So this whole thing simplifies to negative 9 times negative 1 minus 7 times 7. And so this is equal to-- we're in the home stretch-- negative 9 times negative 1. Well, that's positive 9. And then negative 7 times 7 is negative 49. So it's 9 minus 49, which is equal to negative 40. And we are done.