Divide and Simplify:
y^2-25/7 divided by 7y+35/49
To simplify the division of fractions, we can follow these steps:
Step 1: Simplify each fraction individually by factoring, if possible.
In this case, we have two fractions: (y^2 - 25)/7 and (7y + 35)/49.
The first fraction, (y^2 - 25)/7, can be simplified by factoring the numerator as a difference of squares:
(y^2 - 25) = (y + 5)(y - 5).
So, the first fraction becomes ((y + 5)(y - 5))/7.
The second fraction, (7y + 35)/49, can be simplified by factoring out the greatest common factor (GCF) from both the numerator and denominator:
(7y + 35) = 7(y + 5).
So, the second fraction becomes (7(y + 5))/49.
Now, our division problem becomes:
((y + 5)(y - 5))/7 ÷ (7(y + 5))/49.
Step 2: Rewrite the division as a multiplication of the reciprocal of the second fraction.
To do this, we need to invert the second fraction, which means swapping the positions of the numerator and denominator. Since (7(y + 5))/49 becomes 49/(7(y + 5)), our problem becomes:
((y + 5)(y - 5))/7 × 49/(7(y + 5)).
Step 3: Cancel out common factors.
We can cancel out the factors that appear in both the numerator and denominator, if possible. In this case, we can cancel out the factor of (y + 5) in both the numerator and denominator:
[(y + 5)(y - 5)]/(7(y + 5)) × 49/7.
Now, we are left with (y - 5)/1 × 49/7.
Step 4: Multiply across the numerators and denominators.
Multiply the numerators together and the denominators together:
(y - 5) × 49 / (1 × 7).
This simplifies to:
49(y - 5)/7.
So, the simplified expression for (y^2 - 25)/7 ÷ (7y + 35)/49 is 49(y - 5)/7.