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# Systems of equations with graphing: chores

CCSS.Math:

## Video transcript

use graphing to solve the following problem Abby and Ben did house abby invented household chores last weekend together they earned \$50 and Abby earned \$10 more than Ben how much did they each earn so let's define some variables here let's let let's let a equal Abby's earnings Abby's earnings and let's let B equal Ben's earnings Ben's Ben's earnings then they tell us how these earnings relate they first tell us that together they earned \$50 so that statement could be converted in mathematically into well together that means the sum of the two earnings so a plus B needs to be equal to \$50 Abby's Plus Ben's earnings is \$50 and then they tell us Abby earned \$10 more than Ben so we can translate that into Abby is equal to or Abby's earnings is equal to Ben's earnings plus 10 Abby earned \$10 more than Ben so we have a system of two equations and actually with two unknowns and then they say how much did each earn how much did each earn so to do that and they want us to solve this graphically there's multiple ways to solve it but we'll do what they asked us to do let me draw some axes over here and I'll be in the first quadrant since we're only we're dealing with earnings so neither of their earnings can be negative and let me just define the vertical axis as Abby's axis or the Abby's earnings axis and let me define the horizontal axis as Ben's axis or Ben's earnings axis and let me just graph each of these equations and to do that I'm going to take this first equation and I'm going to put it in the equivalent of slope-intercept form it might look a little unfamiliar to you but it really is slope-intercept form let me rewrite it first so we have a plus B is equal to 50 we can subtract B from both sides so let's subtract B from both sides and then we get we get a is equal to negative B plus 50 so if you think about it this way when B is equal to 0 a is going to be 50 so we know our we know our a intercept we could call it we normally we call that it a y-intercept but that is the a axis so this right here let me call this this is 10 20 30 40 and 50 so when if if been made the zero dollars and a B would have to make \$50 based on that first constraint so we know that that's a point on the line right over there and we also know that the slope is negative one that B is the independent variable the way I've written it over here and this coefficient is negative one or another way to think about it is if a is zero then B is going to be 50 if a B made no money then Ben would have to make \$50 and that falls purely out of this equation right here if AB you made nothing then Ben would have to make \$50 so 10 20 30 40 50 so those are those two situations and every point in between will satisfy this first constraint so let me connect the dots so it would look something like that that's due to this first constraint due to the fact that together they earned \$50 now let's think about the second one and Abby earned \$10 more than Ben so that's this equation right here it's really already in our slope intercept form if Ben made zero dollars then Abby would make \$10 so that's our a intercept so it's right over there we could keep doing that our slope is going to be 1 here if Ben makes \$10 then Abby is going to make \$20 if Ben makes \$20 Abby is going to make \$30 then makes we could keep going but I think this gives us the general direction it already hints at a point of intersection so just eyeballing it so we've graphed the two constraints together they are in \$50 that's the magenta constraint right over here Abby earned \$10 more than Ben that's this green constraint right over here and it looks like we have a point of intersection it looks like we have a point of intersection at been earning 20 let me label this is 10 20 30 40 and 50 so this is been earning 20 and a be earning this is 10 20 and 30 and a be earning 30 so just eyeballing it off of this it looks like it looks like a is 30 and B is 20 and let's go verify that make sure that that these levels of earnings for a B and men actually satisfy both constraints so the first constraint is that a B plus men have to make \$50 well 30 plus 20 is \$50 so it meets our first constraint the second constraint is that Abby earned \$10 more than Ben that a B is equal to Ben plus 10 well once again over here Abby is making \$10 more than Ben so it may it meets our second constraint and that's we only have two of them so it meets both of them so that's our solution a beer into \$30 been earned 20