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 here to help. Let's solve these two problems step by step.
Problem 1:
1/(x - 1) - 2/(x^2 - 1) = -1/2
To solve this equation, we first need to find a common denominator for the fractions. The least common denominator (LCD) of (x - 1) and (x^2 - 1) is (x - 1)(x + 1).
Now, we can rewrite the equation with the LCD:
[(x + 1)/(x - 1)(x + 1)] - 2/(x^2 - 1) = -1/2
Next, simplify the fractions:
(x + 1)/(x^2 - 1) - 2/(x^2 - 1) = -1/2
Combine the fractions with the same denominator:
(x + 1 - 2)/(x^2 - 1) = -1/2
Simplify the numerators:
(x - 1)/(x^2 - 1) = -1/2
Now, let's cross-multiply:
2(x - 1) = -1(x^2 - 1)
Expand the equation:
2x - 2 = -x^2 + 1
Rearrange the equation to make it a quadratic equation:
x^2 + 2x - 3 = 0
Next, we'll solve this quadratic equation for x. You can use either factoring, completing the square, or the quadratic formula to find the roots of the equation. I'll use factoring for this example:
(x + 3)(x - 1) = 0
Set each factor equal to zero:
x + 3 = 0 or x - 1 = 0
Solve for x:
x = -3 or x = 1
So, the solutions for this equation are x = -3 and x = 1.
Problem 2:
2/y + (y - 1)/(3y) = 2/5
To solve this equation, we'll find a common denominator for the fractions. The least common denominator (LCD) of y and 3y is 3y.
Rewrite the equation with the LCD:
(2 * 3y)/(y * 3y) + (y - 1)/(3y) = 2/5
Simplify the numerators:
6y/(3y^2) + (y - 1)/(3y) = 2/5
Combine the fractions with the same denominator:
(6y + (y - 1))/(3y) = 2/5
Simplify the numerator:
(7y - 1)/(3y) = 2/5
Now, cross-multiply:
5(7y - 1) = 2(3y)
Expand and simplify:
35y - 5 = 6y
Move the variables to one side of the equation:
35y - 6y = 5
Combine like terms:
29y = 5
Finally, solve for y by dividing both sides of the equation by 29:
y = 5/29
So, the solution for this equation is y = 5/29.