How do you solve #(2y) / (2y+6) + (9y-12) / (3y+9) = (9y+11) / (y+3)#?

Question

Julia Adams · Accepted Answer

Some of these fractions can be simplified.

#(2y)/(2y+6):2/2=y/(y+3)# #(9y-12)/(3y+9):3/3=(3y-4)/(y+3)# So now we have: #y/(y+3)+(3y-4)/(y+3)=(9y+11)/(y+3)# Since the numerators are all equal we may leave them out, on the condition #y!=-3# for this would make the numerators #=0# and the whole equation would be senseless. #y+3y-4=9y+11-># All the #y#'s to one side, all the numbers to the other: #-5y=15->y=-3# And this is exactly the solution that was forbidden. Answer : no sensible solution.

Jameson Allen · Answer

undefined To solve the equation (2y) / (2y+6) + (9y-12) / (3y+9) = (9y+11) / (y+3), we can follow these steps:

1. Simplify the expressions on both sides of the equation.
2. Find a common denominator for all the fractions.
3. Combine the fractions on both sides of the equation.
4. Solve for y by isolating the variable on one side of the equation.
5. Check the solution by substituting the value of y back into the original equation.

Let's go through each step in detail:

1. Simplify the expressions:
   - The expression (2y) / (2y+6) can be simplified to y / (y+3).
   - The expression (9y-12) / (3y+9) can be simplified to 3(y-4) / 3(y+3).
   - The expression (9y+11) / (y+3) remains as it is.

2. Find a common denominator:
   - The common denominator for all the fractions is (y+3)(y+3).

3. Combine the fractions:
   - The equation becomes y / (y+3) + 3(y-4) / 3(y+3) = (9y+11) / (y+3).

4. Solve for y:
   - Multiply both sides of the equation by (y+3) to eliminate the denominators.
   - Simplify the equation and solve for y.

5. Check the solution:
   - Substitute the value of y back into the original equation to ensure it satisfies the equation.

Please let me know if you need further clarification or assistance with any of the steps.