SOLUTION: Solve the equation and check the solution. (Enter your answers as a comma-separated list. Simplify your answers completely. If there is no solution, enter NO SOLUTION.) Solve for y

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Solve the equation and check the solution. (Enter your answers as a comma-separated list. Simplify your answers completely. If there is no solution, enter NO SOLUTION.) Solve for y      Log On


   

Question 1130249: Solve the equation and check the solution. (Enter your answers as a comma-separated list. Simplify your answers completely. If there is no solution, enter NO SOLUTION.) Solve for y.
y + 2/3 = (2y-12)/(3y-9)
Identify any extraneous values. (Enter your answers as a comma-separated list. Simplify your answers completely. If there are no extraneous values, enter NO SOLUTION.)
Found 2 solutions by josgarithmetic, MathLover1:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
Try seeing the equation with some factorizations done:

y%2B2%2F3=%282%28y-6%29%29%2F%283%28y-3%29%29

The simplest common denominator can be identified as 3%28y-3%29.

Multiply entire left and right sides of the equation by 3(y-3), which allows to clear denominators; and continue to simplify and solve.

See similar exercises already responded to .

https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1130241.html
https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1130241.html

https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1130238.html
https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1130238.html

https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1130234.html
https://www.algebra.com/algebra/homework/Polynomials-and-rational-expressions/Polynomials-and-rational-expressions.faq.question.1130234.html

Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
y+%2B+2%2F3+=+%282y-12%29%2F%283y-9%29
%283y+%2B+2%29%2F3+=+%282y-12%29%2F%283y-9%29............cross multiply
%283y+%2B+2%29%283y-9%29+=+3%282y-12%29
9y%5E2-27y%2B6y-18+=+6y-36
9y%5E2-27y%2B6y-18+-+6y%2B36=0
9y%5E2-27y%2Bcross%286y%29-18+-cross%28+6y%29%2B36=0
9y%5E2-27y%2B18=0......simplify, both sides divide by 9
y%5E2-3y%2B2=0..............factor completely
y%5E2-y-2y%2B2=0
%28y%5E2-y%29-%282y-2%29=0
y%28y-1%29-2%28y-1%29=0
%28y+-+2%29+%28y+-+1%29+=+0
solutions:
if %28y+-+2%29++=+0->y=2
if +%28y+-+1%29+=+0 ->y=1



check the solution:

y+%2B+2%2F3+=+%282y-12%29%2F%283y-9%29->y=2
2+%2B+2%2F3+=+%282%2A2-12%29%2F%283%2A2-9%29
6%2F3+%2B+2%2F3+=+%284-12%29%2F%286-9%29
8%2F3+=+-8%2F-3
8%2F3+=+8%2F3-> true; so, y=2 is a solution

y+%2B+2%2F3+=+%282y-12%29%2F%283y-9%29->y=1
1+%2B+2%2F3+=+%282%2A1-12%29%2F%283%2A1-9%29
3%2F3+%2B+2%2F3+=+%282-12%29%2F%283-9%29
5%2F3+=+-10%2F-6
5%2F3+=+-5%2F-3
5%2F3+=+5%2F3-> true; so, y=1 is a solution

there are no extraneous values
your answer: 2, 1