Problem:
(y+2)/(y) = 1/(y-5)
Directions:
Solve the rational expression.
Remember that if:
a/b = c/d
then:
a * d = b * c
I believe in school they call this the fraction crossover rule or fraction cross rule or something like that.
i would work it like a proportion and cross multiply and then divide.
(y+2)/(y) = 1/(y-5)
(y-5)(y+2) = y
y^2-3y-10 = y
y^2-4y-10 = 0
Since there are no factors of 10 whose difference is 4, use the quadratic formula to solve.
To solve the rational expression (y+2)/(y) = 1/(y-5), you can follow the steps below:
Step 1: Remember the fraction crossover rule, which states that if a/b = c/d, then a * d = b * c.
Step 2: Apply the fraction crossover rule to the given equation. Multiply the numerator of the first fraction, (y+2), with the denominator of the second fraction, (y-5), and set it equal to the product of the denominator of the first fraction, y, with the numerator of the second fraction, 1:
(y+2)(y-5) = y(1)
Step 3: Simplify the expression:
(y^2 - 3y - 10) = y
Step 4: Rearrange the equation and set it equal to zero:
y^2 - 4y - 10 = 0
Step 5: Since there are no factors of 10 whose difference is 4, you need to use the quadratic formula to solve the equation.
The quadratic formula is given by:
y = (-b ± √(b^2 - 4ac)) / 2a
For our equation y^2 - 4y - 10 = 0:
a = 1, b = -4, c = -10
Substituting the values into the quadratic formula:
y = (-(-4) ± √((-4)^2 - 4*1*(-10))) / (2*1)
y = (4 ± √(16 + 40)) / 2
y = (4 ± √56) / 2
y = (4 ± 2√14) / 2
Step 6: Simplify further:
y = 2 ± √14
Therefore, the solution to the rational expression is:
y = 2 + √14 or y = 2 - √14.