Simplify:
y - 3 / y - 2 - y + 1 / 2y - 5 + -4y + 7 / 2y^2 - 9y + 10
I'm sure there are some missing parentheses. Making a guess, I'd say you meant
(y-3)/(y-2) - (y+1)/(2y-5) + (-4y+7)/(2y^2-9y+10)
putting all over a common denominator, we have
[(y-3)(2y-5) - (y+1)(y-2) + (-4y+7)]/(2y^2-9y+10)
((2y^2-11y+15) - (y^2-y-2) + (-4y+7))/(2y^2-9y+10)
= (y^2-14y+24)/(2y^2-9y+10)
= (y-2)(y-12)/[(y-2)(2y-5)]
= (y-12)/(2y-5)
Thank you very much. Yeah, I should've added the parenthesis to make it look familiar.
To simplify the expression (y - 3) / (y - 2) - (y + 1) / (2y - 5) + (-4y + 7) / (2y^2 - 9y + 10), we need to find a common denominator for all fractions and then combine them.
1. Start with the first two fractions: (y - 3) / (y - 2) - (y + 1) / (2y - 5)
2. To find the common denominator for these fractions, we need to factor the denominators:
- For the first fraction, the denominator (y - 2) is already factored.
- For the second fraction, the denominator (2y - 5) is already factored.
3. Now, multiply the numerators and denominators of the first fraction by the denominator of the second fraction and vice versa:
(y - 3)(2y - 5) / [(y - 2)(2y - 5)] - (y + 1)(y - 2) / [(y - 2)(2y - 5)]
4. Distribute and simplify each fraction:
[(2y^2 - 5y) - (y^2 - 3y - 2)] / [(y - 2)(2y - 5)]
Simplifying further:
[2y^2 - 5y - y^2 + 3y + 2] / [(y - 2)(2y - 5)]
Combining like terms:
(y^2 - 2y + 2) / [(y - 2)(2y - 5)]
5. Now, let's work with the third fraction: (-4y + 7) / (2y^2 - 9y + 10).
6. Similar to the previous step, we need to factor the denominator:
2y^2 - 9y + 10 = (2y - 5)(y - 2)
7. Now, rewrite the fraction with the common denominator:
(-4y + 7)(y - 2) / [(2y - 5)(y - 2)]
8. Simplify the numerator:
(-4y^2 + 8y + 7y - 14) / [(2y - 5)(y - 2)]
Simplifying further:
(-4y^2 + 15y - 14) / [(2y - 5)(y - 2)]
9. Now, we can combine all three terms by adding or subtracting them:
(y^2 - 2y + 2) / [(y - 2)(2y - 5)] - (-4y^2 + 15y - 14) / [(2y - 5)(y - 2)]
10. Since we have the same denominator for both terms, we can combine them:
[(y^2 - 2y + 2) - (-4y^2 + 15y - 14)] / [(2y - 5)(y - 2)]
Simplifying the numerator:
(y^2 - 2y + 2 + 4y^2 - 15y + 14) / [(2y - 5)(y - 2)]
Combining like terms:
(3y^2 - 17y + 16) / [(2y - 5)(y - 2)]
So, the simplified expression is (3y^2 - 17y + 16) / [(2y - 5)(y - 2)].