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# Systems of equations with graphing: chores

Videotranscript

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.