I appreciate all of your help. We had about 77 of these problems to do. These are the LAST two problems that I need some help on. Thank you so much for everything.
1/x-1 minus 2/x^2-1= -1/2
2/y plus y-1/3y= 2/5
You're welcome! I'm glad I could help. Let's start by solving the first problem:
1. Identify the denominators in the equation: x-1 and x^2-1.
2. To simplify the equation, let's find the common denominator for the fractions. Note that x^2-1 is a difference of squares, which can be factored into (x-1)(x+1).
3. Rewrite the fractions with the common denominator:
For the first term, 1/x-1, the common denominator is (x-1)(x+1), so we multiply the numerator and denominator by (x+1):
(1 * (x+1))/(x-1)(x+1) = (x+1)/((x-1)(x+1)).
For the second term, 2/x^2-1, the common denominator is (x-1)(x+1), so we multiply the numerator and denominator by 2:
(2 * 2)/((x-1)(x+1)) = 4/((x-1)(x+1)).
4. Now we can rewrite the equation with the common denominator:
(x+1)/((x-1)(x+1)) - 4/((x-1)(x+1)) = -1/2.
5. Combine the fractions on the left side of the equation:
(x+1 - 4)/((x-1)(x+1)) = -1/2.
(x - 3)/((x-1)(x+1)) = -1/2.
6. Cross-multiply to eliminate the fractions:
2(x - 3) = -1((x-1)(x+1)).
2x - 6 = -x^2 + 1.
7. Rearrange the equation to bring it to standard form:
x^2 + 2x - 7 = 0.
8. Solve the quadratic equation either by factoring, completing the square, or using the quadratic formula.
By factoring, we find that (x - 1)(x + 3) = 0.
So x - 1 = 0 or x + 3 = 0.
Therefore, x = 1 or x = -3 are the solutions to the equation.
Now, let's move on to the second problem:
1. Identify the denominators in the equation: y and 3y.
2. To simplify the equation, let's find the common denominator for the fractions. The least common multiple of y and 3y is 3y.
3. Rewrite the fractions with the common denominator:
For the first term, 2/5, we need to multiply the numerator and denominator by (3y/y):
(2 * 3y)/(5 * 3y) = (6y)/(15y).
For the second term, y-1/3y, we don't need to make any changes.
4. Now we can rewrite the equation with the common denominator:
(6y)/(15y) + (y-1)/(3y) = 2/5.
5. Combine the fractions on the left side of the equation:
(6y + (y-1))/(15y) = 2/5.
(7y - 1)/(15y) = 2/5.
6. Cross-multiply to eliminate the fractions:
5(7y - 1) = 2(15y).
35y - 5 = 30y.
7. Rearrange the equation to bring it to standard form:
35y - 30y = 5.
5y = 5.
8. Solve for y:
y = 1 is the solution to the equation.
I hope this helps you with the last two problems! If you have any further questions, feel free to ask.