Final answer:
To solve the given rational equation, we need to eliminate the fractions by multiplying both sides of the equation by the denominators. Then we can simplify and factor the resulting quadratic equation to find the value of y.
Step-by-step explanation:
To solve the rational equation,4+(1)/(y+2)=-(7)/(y-1) we need to eliminate the fractions first. Multiplying both sides of the equation by the denominators (y+2) and (y-1) will give us a quadratic equation. Then we can solve for y by simplifying and factoring the quadratic equation.
First, multiply both sides by (y+2) and (y-1) to eliminate the fractions:
4(y-1) + (y+2) = -7(y+2)
Next, distribute and simplify:
4y - 4 + y + 2 = -7y -14
Combine like terms:
5y - 2 = -7y - 14
Add 7y to both sides:
12y - 2 = -14
Add 2 to both sides:
12y = -12
Divide both sides by 12:
y = -1
So the solution to the equation is y = -1.