# Solve for y 9+2/(y-1)=-4/(y-2)

9+2y-1=-4y-2
Subtract 9 from both sides of the equation.
2y-1=-4y-2-9
Find the LCD of the terms in the equation.
Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.
y-1,y-2,1
The LCM is the smallest positive number that all of the numbers divide into evenly.
1. List the prime factors of each number.
2. Multiply each factor the greatest number of times it occurs in either number.
The number 1 is not a prime number because it only has one positive factor, which is itself.
Not prime
The LCM of 1,1,1 is the result of multiplying all prime factors the greatest number of times they occur in either number.
1
The factor for y-1 is y-1 itself.
(y-1)=y-1
(y-1) occurs 1 time.
The factor for y-2 is y-2 itself.
(y-2)=y-2
(y-2) occurs 1 time.
The LCM of y-1,y-2 is the result of multiplying all factors the greatest number of times they occur in either term.
(y-1)(y-2)
(y-1)(y-2)
Multiply each term by (y-1)(y-2) and simplify.
Multiply each term in 2y-1=-4y-2-9 by (y-1)(y-2) in order to remove all the denominators from the equation.
2y-1⋅((y-1)(y-2))=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
Simplify 2y-1⋅((y-1)(y-2)).
Cancel the common factor of y-1.
Cancel the common factor.
2y-1⋅((y-1)(y-2))=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
Rewrite the expression.
2⋅(y-2)=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
2⋅(y-2)=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
Apply the distributive property.
2y+2⋅-2=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
Multiply 2 by -2.
2y-4=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
2y-4=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
Simplify -4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2)).
Simplify each term.
Cancel the common factor of y-2.
Move the leading negative in -4y-2 into the numerator.
2y-4=-4y-2⋅((y-1)(y-2))-9⋅((y-1)(y-2))
Factor y-2 out of (y-1)(y-2).
2y-4=-4y-2⋅((y-2)(y-1))-9⋅((y-1)(y-2))
Cancel the common factor.
2y-4=-4y-2⋅((y-2)(y-1))-9⋅((y-1)(y-2))
Rewrite the expression.
2y-4=-4⋅(y-1)-9⋅((y-1)(y-2))
2y-4=-4⋅(y-1)-9⋅((y-1)(y-2))
Apply the distributive property.
2y-4=-4y-4⋅-1-9⋅((y-1)(y-2))
Multiply -4 by -1.
2y-4=-4y+4-9⋅((y-1)(y-2))
Expand (y-1)(y-2) using the FOIL Method.
Apply the distributive property.
2y-4=-4y+4-9⋅(y(y-2)-1(y-2))
Apply the distributive property.
2y-4=-4y+4-9⋅(y⋅y+y⋅-2-1(y-2))
Apply the distributive property.
2y-4=-4y+4-9⋅(y⋅y+y⋅-2-1y-1⋅-2)
2y-4=-4y+4-9⋅(y⋅y+y⋅-2-1y-1⋅-2)
Simplify and combine like terms.
Simplify each term.
Multiply y by y.
2y-4=-4y+4-9⋅(y2+y⋅-2-1y-1⋅-2)
Move -2 to the left of y.
2y-4=-4y+4-9⋅(y2-2⋅y-1y-1⋅-2)
Rewrite -1y as -y.
2y-4=-4y+4-9⋅(y2-2y-y-1⋅-2)
Multiply -1 by -2.
2y-4=-4y+4-9⋅(y2-2y-y+2)
2y-4=-4y+4-9⋅(y2-2y-y+2)
Subtract y from -2y.
2y-4=-4y+4-9⋅(y2-3y+2)
2y-4=-4y+4-9⋅(y2-3y+2)
Apply the distributive property.
2y-4=-4y+4-9y2-9(-3y)-9⋅2
Simplify.
Multiply -3 by -9.
2y-4=-4y+4-9y2+27y-9⋅2
Multiply -9 by 2.
2y-4=-4y+4-9y2+27y-18
2y-4=-4y+4-9y2+27y-18
2y-4=-4y+4-9y2+27y-18
2y-4=23y+4-9y2-18
Subtract 18 from 4.
2y-4=23y-9y2-14
2y-4=23y-9y2-14
2y-4=23y-9y2-14
2y-4=23y-9y2-14
Solve the equation.
Since y is on the right side of the equation, switch the sides so it is on the left side of the equation.
23y-9y2-14=2y-4
Set the equation equal to zero.
Move all the expressions to the left side of the equation.
Move 2y to the left side of the equation by subtracting it from both sides.
23y-9y2-14-2y=-4
Move 4 to the left side of the equation by adding it to both sides.
23y-9y2-14-2y+4=0
23y-9y2-14-2y+4=0
Simplify 23y-9y2-14-2y+4.
Subtract 2y from 23y.
21y-9y2-14+4=0
21y-9y2-10=0
21y-9y2-10=0
21y-9y2-10=0
Factor the left side of the equation.
Let u=y. Substitute u for all occurrences of y.
21u-9u2-10
Factor by grouping.
Reorder terms.
-9u2+21u-10
For a polynomial of the form ax2+bx+c, rewrite the middle term as a sum of two terms whose product is a⋅c=-9⋅-10=90 and whose sum is b=21.
Factor 21 out of 21u.
-9u2+21(u)-10
Rewrite 21 as 6 plus 15
-9u2+(6+15)u-10
Apply the distributive property.
-9u2+6u+15u-10
-9u2+6u+15u-10
Factor out the greatest common factor from each group.
Group the first two terms and the last two terms.
(-9u2+6u)+15u-10
Factor out the greatest common factor (GCF) from each group.
3u(-3u+2)-5(-3u+2)
3u(-3u+2)-5(-3u+2)
Factor the polynomial by factoring out the greatest common factor, -3u+2.
(-3u+2)(3u-5)
(-3u+2)(3u-5)
Replace all occurrences of u with y.
(-3y+2)(3y-5)
Replace the left side with the factored expression.
(-3y+2)(3y-5)=0
(-3y+2)(3y-5)=0
If any individual factor on the left side of the equation is equal to 0, the entire expression will be equal to 0.
-3y+2=0
3y-5=0
Set the first factor equal to 0 and solve.
Set the first factor equal to 0.
-3y+2=0
Subtract 2 from both sides of the equation.
-3y=-2
Divide each term by -3 and simplify.
Divide each term in -3y=-2 by -3.
-3y-3=-2-3
Cancel the common factor of -3.
Cancel the common factor.
-3y-3=-2-3
Divide y by 1.
y=-2-3
y=-2-3
Dividing two negative values results in a positive value.
y=23
y=23
y=23
Set the next factor equal to 0 and solve.
Set the next factor equal to 0.
3y-5=0
Add 5 to both sides of the equation.
3y=5
Divide each term by 3 and simplify.
Divide each term in 3y=5 by 3.
3y3=53
Cancel the common factor of 3.
Cancel the common factor.
3y3=53
Divide y by 1.
y=53
y=53
y=53
y=53
The final solution is all the values that make (-3y+2)(3y-5)=0 true.
y=23,53
y=23,53
The result can be shown in multiple forms.
Exact Form:
y=23,53
Decimal Form:
y=0.6‾,1.6‾
Mixed Number Form:
y=23,123
Solve for y 9+2/(y-1)=-4/(y-2)