Systems of equations with graphing
Use graphing to solve the following problem. Abby and Ben did 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 A equal Abby's earnings. And let's let B equal Ben's earnings. Then they tell us how these earnings relate. They first tell us that together they earned $50. So that statement can be converted 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 could translate that into 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? 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 ask us to do. Let me draw some axes over here. And I'll be in the first quadrant since 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 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 A intercept, we could call it. We normally would call that a y-intercept, but now this is the A axis. So this right here, let me call this 10, 20, 30, 40, and 50. So if Ben made $0, then Abby 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 1, that B is the independent variable, the way I've written it over here, and this coefficient is negative 1. Or another way to think about it is if A is 0, then B is going to be 50. If Abby made no money, then Ben would have to make $50. And that falls purely out of this equation right here. If Abby 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. Abby earned $10 more than Ben. So that's this equation right here. And it's really already in our slope-intercept form. If Ben made $0, 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's going to make $20. If Ben makes $20, Abby's going to make $30. 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 to the two constraints, together they earned $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. And it looks like we have a point of intersection at Ben earning $20. Let me label this as 10, 20, 30, 40, and 50. So this is Ben earning $20 and Abby earning-- this is 10, 20, and 30. And Abby earning $30. So just eyeballing it off of this, it looks like A is 30 and B is 20. And let's go verify, make sure that these levels of earnings for Abby and Ben actually satisfy both constraints. So the first constraint is that Abby plus Ben 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 Abby is equal to Ben plus 10. Well, once again, over here, Abby is making $10 more than Ben. So it meets our second constraint. And we only have two of them. So it meets both of them. So that's our solution. Abby earned $30. Ben earned $20.