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

Current time:0:00Total duration:4:39

CCSS.Math:

I'm not 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 5 a plus 5b plus 5c equal 2 and the first reaction is 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 you don't 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 five 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 that first equation gave us all the information that we needed they told us that a plus B plus C is equal to 7 a plus B plus C is equal to 7 so 5 times a plus B plus C is the same exact thing and I think you see where this is going now is 5 times 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 one and we have two more variables here we're told that X plus y is equal to seven and then they asked us this big hairy thing what is this equal to and I'll give you a few seconds to give it give a little bit of thought well you could imagine you've given three variables with one equation up here and our 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 we might want to do is rearrange it so that we have the X's and Y's kind of group kind of grouped together and the A's B's and C's kind of group 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 9 a negative 9 B I'll go and order alphabetical order negative 9 B and we have negative 9 C and I think you see what's emerging negative 9 C 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 minus 7y so all I have done is rearranged this expression here but this makes it in a way or it makes it a little bit clearer of what's going on here this these first 3 terms 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 a 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 this and 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 2 well they tell us right over here X plus y is equal is equal to 7 so this whole thing simplifies to negative 9 times negative 1 times negative 1 minus 7 minus 7 times 7 times 7 and so this is equal to or the homestretch negative 9 times negative 1 well that's positive 9 and then negative seven times seven is negative 49 so it's 9 minus 49 which is equal to negative 40 and we are done