Abby and Ben did household chores last weekend.
Together they earned $50.
That looks like an interesting statement.
Let me underline that.
Together they earned $50 and Abby earned $10 more than Ben.
So they also tell us--let me underline that-- Abby earned
$10 more than Ben.
How much did each earn?
So that's the first question that we have to answer.
Then they say, if you were to use the graph above, and I put
it on the side here, which strategy is most useful in
helping you find the answer to the problem?
So there's kind of a meta-level question of how do
we answer that first question.
Find the equation for the blue line?
Red's y-intercept?
Find the points of intersection?
Fair enough.
So let's just think about that first question and then we can
answer this later, depending on how we
answered that first question.
So that first statement, together they earned $50.
What does that tell us?
Well, if we say that Abby's earnings are represented by a,
and Ben's earnings are represented by b, that tells
us that a plus b must be $50.
So a plus b have to be equal to $50.
Abby's earnings plus Ben's earnings are $50.
Together they earned $50.
That's all that statement is telling us.
And if we see how they graphed things here, on the vertical
axis, what we normally associate with the y-axis,
they graphed Abby's earnings.
This is Abby's earnings right there.
So we could say this is the a-axis.
And then on the horizontal axis, what we normally
associate with the x-axis, they've charted Ben's
earnings, so we could say that this is the b-axis.
So if we were to just map this into the traditional
slope-intercept form, we don't have x's and y's here.
We have b's and a's.
So we just want to express a in terms of b, the vertical
variable in terms of the horizontal variable.
That's just like the y equals mx plus b form.
And the easiest way to do that is just subtract b from both
sides of this equation.
And we get a is equal to negative b plus 50.
So in this situation, you have a slope of negative 1.
Let me write this down.
Slope of negative 1, that's what's multiplying the
independent variable, and then you have a y-intercept of 50,
or maybe we should say an a-intercept.
But I'll write y-intercept because that's what we're
familiar with.
You can view this as also like the y or the vertical axis.
y-intercept of 50.
And you can see they graphed that.
You have an a-intercept or y-intercept of 50, and the
slope is negative 1.
If we move 20 in the positive x-direction, we are moving
down 20, or we're going 20 in the negative vertical
direction, or in the negative a-direction.
If we go 50 in the positive-- I shouldn't say x-direction--
the positive b-direction, we go down 50 in the a-direction,
so the slope is negative 1.
So this red line right here is the graph that depicts this
first constraint.
It's important to realize that, look, all this graph
shows is all the different possible combinations of
Abby's and Ben's earnings given only this first
constraint.
Maybe Abby made $50 and Ben earned $0.
Maybe Ben earned $50 and Abby earned $0.
Maybe they both earned $25.
Everything on this line would be a valid combination of
Abby's and Ben's earnings given the constraint that they
earned $50 combined.
Now, the second statement.
Abby earned $10 more than Ben.
So if we were to write that, that's saying that Abby is
equal to Ben's earnings plus $10.
Abby earned $10 more than Ben.
And here the slope is 1, you see that right there.
There's an implicit 1 multiplying times this b.
And then the y-intercept is 10.
And they graphed that here.
The y-intercept is 10 and the slope is 1.
You move 10 in the positive b-direction, you move up 10 in
the positive a-direction.
So this blue line represents all of the possible
combinations of Abby and Ben's earnings if you only had this
constraint that Abby earned $10 more than Ben.
If Ben earns $0, Abby earns $10.
If Ben earns $10, Abby earns $20.
If Ben earns $60, Abby earns $70.
Now, we aren't dealing with only the blue constraint or
only the red constraint.
We're dealing with both of them.
We know that both of them are true.
So whatever earnings Abby and Ben made, they have to satisfy
both constraints or they have to sit on both of these lines.
Now, what is the only point that is sitting on both of
these lines?
Well, the only point that sits on both lines is the
intersection of those lines.
That point right there is a valid combination from the
point of view of the blue constraint that Abby earned
$10 more than Ben.
And it's also a valid point or a valid combination from the
point of view of the red constraint that Abby and Ben
earned $50.
So that point represents Abby's and Ben's earnings that
satisfy both constraints.
And we can eyeball it right here.
It looks like Abby earned $30 and Ben earned $20.
And that makes sense.
Combined, $20 plus $30 or $30 plus $20 is $50, so we meet
our first constraint.
And we see that Abby, $30, made $10 more than Ben, who
only made $20.
So if you look at the point of intersection, you will have
solved the problem.
Abby made $30, Ben earned $20.
So answering this second-- so we know the answer-- so Abby
earned $30 and Ben earned $20.
Now, answering the second-level, meta-level
question, which is the best strategy?
We see, well, we had to find the points of intersection, so
that is choice C.
We had to find the lines' point of intersection.
Finding these other things really wouldn't have helped us
solve the problem.