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CCSS.Math: ,

so this is an example from the Khan Academy exercise graphing solutions to two variable linear equations and they tell us to complete the table so each row represents a solution of the following equation and they give us the equation and then they want us to figure out what does y equal when X is equal to negative five and what is X equal and Y is equal to eight and to figure this out I've actually copy and pasted this part of the problem onto my scratch pad so let me get let me get that out and so this is the exact same problem there's a couple of ways that we could try to tackle it one way is you could try to simplify this more get all your X's on one side and all your Y's on the other side or we could just literally substitute when x equals negative 5 what happened what must y equal actually let me do it the second way first so if we take this equation and we substitute X with negative 5 what do we get we get negative 3 times well we're going to say X is negative 5 times negative 5 plus 7y is equal to 5 times X is once again it's going to be negative 5 X is negative 5 5 times negative 5 plus 2y see negative 3 times negative 5 is positive 15 plus 7y is equal to negative 25 plus 2y and now to solve for y let's see i could subtract 2y from both sides so that i get rid of the 2i here on the right so let me subtract 2y subtract 2y from both sides and then if I want all my constants on the right hand side I can subtract 15 from both sides so let me subtract 15 from both sides and I am going to be left with 15 minus 15 that's 0 that's the whole point of subtracting 15 from both sides so I get rid of this 15 here 7y minus 2i 7 of something minus 2 of that of that same something is going to be 5 of that something it's going to be equal to 5y is equal to negative 25 minus 15 well that's going to be negative 40 and then 2i minus 2i that's just going to be 0 that was the whole point of subtracting 2y from both sides so you have 5 times y is equal to negative or tea or if we divide both sides by five we divide both sides by five we would get Y is equal to negative eight so when X is equal to negative five y is equal to negative eight y is equal to negative eight and actually we can fill that in so this Y is going to be equal to negative eight and now we've got to figure this out what does X equal when y is positive eight well we can go back to our scratch pad here and now let's take the same equation but let's make y equal to positive eight so you have negative 3x plus seven now Y is going to be eight Y is eight seven times eight is equal to five times X plus two x once again Y is eight two times eight so we get negative three X plus 56 that's fifty six is equal to five x plus sixteen now if we want to get all of our constants on one side and of all of our X terms on the other side well that's what what could we do let's see we could add 3x to both sides that would put that would get rid of all of the X's on this side and put them all on this side so we're going to add 3x to both sides and let's see if we want to get all the constants on the left-hand side we want to get rid of the 16 so we could subtract 16 from the right-hand side if we do it from the right we're going to have to do it from the left as well and we're going to be left with these cancel out 56 minus 16 is positive 40 and then let's see 16 minus 16 is 0 5 X plus 3x is equal to 8x we get 8x is equal to 40 we could divide both sides by 8 and we get 5 is equal to X so this right over here is going to be equal to 5 so let's go back let's go back now so when y is positive 8 X is positive 5 now they asked us use your two solutions to graph the equation so let's see if we can do a whoops let me let me use my mouse now so to graph the equation so what x is negative five y is negative eight so the point negative five comma negative eight so that's right over there so let me move my browser up so you can see that negative five when X is negative five y is negative eight and when X is positive five we see that up here when X is positive five y is positive eight when X is positive five y is positive eight and we're done we could check our answer if we like we got it right now I said there was two ways to tackle it I kind of just did it I guess you could say the naive way I just substituted negative five directly into this and solve for y and then I substituted y equals positive eight directly into this and then solved for X another way that I could have done it that actually probably would have been weird for sure would have been the easier way to do it is ahead of time to try to simplify this expression so what I could have done right from the get-go said hey let's put all my X's on one side and all my Y's on the other side so this is negative three X plus seven Y is equal to five x plus two Y and let's say I want to get all my Y's on the left and all my X's on the right so I don't want this negative three X on the left so I'd want to add 3x adding 3x would cancel this out but I can't just do it on the left hand side I have to do it on the right hand side as well and then if I want to get rid of this two Y on the right I could subtract 2y from the right but of course I'd also want to do it from the left and then what am I left with so negative 3x plus 3x is 0 7y minus 2y is 5y and then i have 5x plus 3x is 8 x 2i minus 2y is 0 and then if i wanted to i could solve for y i could divide both sides by 5 and get y is equal to 8/5 X so this right over here is represents the same exact equation is this over here it's just written in a different way all of the XY pairs that satisfy this would satisfy this and vice versa and this is much easier because if X is now negative 5 if is negative 5y would be eight fifths times negative five well that's going to be negative eight and when y is equal to eight well you actually could even do this up here you could say five times 8 is equal to 8x and then you can see well five times eight is the same thing as 8 times 5 so X would be equal to five so I think this would actually been a simpler way to do it you see it I was able to do the entire problem this little white space here instead of having to do all this slightly slightly hairier algebra