{y-2}/(a^2-5a+6)=(a+1)/(a-3)-(a+2)/(a-2)
How can I solve for y in terms of the other variables???
I tried many times, but I just get y=6,can't express y in terms of variables.
Can anyone please correct me or teach me?THANKS A LOT!
The denominator of the left side factors to
(a-2)(a-3) which just happen to be found as denominators on the right side.
so...
if we multiply both sides by (a-2)(a-3) we get
y - 2 = (a+1)(a-2) - (a+2)(a-3)
expand the right sided, take the -2 over to the right side, and collect your like terms.
You are done!
that means that still equal to 6 right?
gee, I guess you are right!
both the a^2 and the a term drop out.
I should have finished my solution, :)
y = 6 !!!!
however,6 is not a variable..the question is asking to solve for y in terms of other variables
why don't you state in your solution that the variables canceled out?
Trust your math!
In this case you were not able to solve in terms of the other variable.
In other words y was not a function of a
I suppose one could get silly and write
y = 6 + 0a
To solve for y in terms of the other variables, we need to simplify the given equation and isolate the variable y on one side of the equation. Let's go step-by-step:
1. Start by simplifying the expression on the right side of the equation: (a+1)/(a-3) - (a+2)/(a-2)
To combine the fractions with different denominators, we need to find the least common denominator (LCD), which in this case is (a-3)(a-2).
So the expression becomes: [(a+1)(a-2) - (a+2)(a-3)] / [(a-3)(a-2)]
Expanding the brackets, we get: [(a^2 - a - 2) - (a^2 - a - 2a - 6)] / [(a-3)(a-2)]
Simplifying further, we have: [a^2 - a - 2 - a^2 + a + 2a + 6] / [(a-3)(a-2)]
Combining like terms: [3a + 4] / [(a-3)(a-2)]
2. Now our equation becomes: (y - 2) / (a^2 - 5a + 6) = [3a + 4] / [(a-3)(a-2)]
3. Cross-multiply to get rid of the denominators.
(y - 2) * [(a-3)(a-2)] = [3a + 4] * (a^2 - 5a + 6)
Expanding both sides of the equation:
(y - 2) * (a^2 - 5a + 6 - 5a + 25 - 30) = (3a + 4) * (a^2 - 5a + 6)
Simplifying further:
(y - 2) * (a^2 - 10a + 1) = (3a + 4) * (a^2 - 5a + 6)
4. Distribute and simplify the equation:
y(a^2 - 10a + 1) - 2(a^2 - 10a + 1) = (3a + 4)(a^2 - 5a + 6)
y(a^2) - y(10a) + y(1) - 2(a^2) + 2(10a) - 2(1) = (3a)(a^2) + (3a)(-5a) + (3a)(6) + 4(a^2) - 4(5a) + 4(6)
Now, gather the like terms on each side:
(a^2)y - 10(a)y + y - 2(a^2) + 20(a) - 2 = 3(a^3) - 15(a^2) + 18(a) + 4(a^2) - 20(a) + 24
Combining the like terms:
a^2y - 10ay + y - 2a^2 + 20a - 2 = 3a^3 - 11a^2 - 2a + 24
5. Rearrange the equation to isolate the variable y:
a^2y - 10ay + y - 2a^2 + 20a - 2 - 3a^3 + 11a^2 + 2a - 24 = 0
Combine like terms:
a^2y - 10ay + y - 2a^2 + 11a^2 + 20a + 2a - 3a^3 - 24 = 0
-3a^3 + a^2y - y + 29a - 2 = 0
At this point, the equation cannot be solved explicitly for y in terms of a, as it is a cubic equation involving the variable y. Therefore, it is not possible to express y solely in terms of the other variables. The expression y = 6 you obtained earlier may not be correct.