Welcome to a practice SAT math question
from the Algebra and Functions section.
If x does not equal two y,
then we want to find the sum
of these two rational expressions.
We're told x does not equal two y.
Because if it did, we'd have
zeros in the denominator,
meaning division by zero, which
would make this undefined.
There are a couple ways to approach this,
but I think it's going to be helpful
if we rewrite each fraction
so that the x terms
are first and the y terms are second.
Meaning for this first fraction,
we'll leave the numerator the same,
we have x minus two y
divided by the quantity
negative x plus two y.
Notice how we changed the order,
but the x is still negative
and the two y is still positive.
And then for the second fraction,
we'll change the order of the numerator,
so we'll have negative x plus two y
and our denominator is going
to stay the same x minus two y.
Now if we take a look at our denominators,
notice how they look almost the same,
but their actually opposites.
This is a negative x,
this is a positive x.
This is a positive two y,
this is a negative two y.
Which means if we were to
multiply this denominator
by negative one,
we would have our common
denominator which we need
in order to add fractions.
But if we multiply the
denominator by negative one,
we also have to multiply the
numerator by negative one.
So we're going to multiply
this first fraction by
negative one over negative one.
So again we'll multiply
this first fraction
by negative one over negative one.
When we do this, looking at the numerator,
we would have negative x plus two y.
And the denominator would
be positive x minus two y.
Which now notice, gives
us our common denominator,
which we need in order
to add these fractions.
So to add these two fractions,
the denominator is going to stay the same,
and we'll add the numerators.
So negative x plus negative
x is negative two x.
And two y plus two y is
four y, so plus four y.
But now notice how we can simplify this.
We'll start by factoring the numerator.
Notice how these two terms
have a common factor of two.
But let's go ahead and
factor out negative two.
So we have negative two times a quantity
positive x minus two y.
Notice once we have this factored,
we have a common factor of x minus two y
in the numerator and denominator
which would simplify to one
leaving us with a sum of negative two.
So our answer is [e], negative two.
But let's also take a
look at a second method
for finding this sum.
Instead of multiplying this first fraction
by negative one over negative one,
let's say we factor out
negative one instead.
If we factor out negative
one from this denominator,
the numerator would stay the same.
So we would have x minus two y.
And we'll factor out a negative one
which is going to change
the sign of these two terms.
So we'll have positive x minus two y.
And for the second
fraction, we'll factor out
negative one from the numerator.
So we'd have negative
one times the quantity
positive x minus two y.
And our denominator stays x minus two y.
Notice in this form,
each fraction simplifies.
To help us recognize how
this is going to simplify,
it might be helpful to put
a one here and a one here;
which of course would not change anything.
Since we're multiplying by a positive one.
But notice how for this first fraction,
we have the quantity x minus two over
the quantity x minus two,
which simplifies to one.
And the same in the second fraction.
So for the first fraction,
we're left with positive
one divided by negative one,
which is negative one.
Plus the second fraction we have
negative one divided by positive one
which is also negative one.
Giving us a sum of negative two.
So whichever method we use,
the sum is negative two.
I hope you found this helpful.
