1. Subtract and simplify
18/y-8 - 18/8-y
2. subtract simplify if possible
v-5/v - 7v-31/11v
3. Simplify by removing factors of 1
r^2-81/(r+9)^2
4. Divide and simplify
v^2-16/25v+100 divided by v-4/20
5. find all rational numbers for which the rational expression is undefined:
-18/7w
6. solve:
2/Z+4= 1/z-6
1. 18 / (y - 8) - 18 / (8 - y),
Multiply numerator and denominator of
2nd fraction by -1 and get:
18 / (y - 8) + 18 / (y - 8) =
36 / (y - 8).
2. (v - 5) / v - (7v - 31) / 11v,
Common denominator = 11v:
(11v - 55 -7v + 31) / 11v,
Combine like-terms:
(4v - 24) / 11v = 4(v - 6) / 11v.
3. (r^2 - 81) / (r + 9)^2,
Factor the numerator:
(r + 9) (r - 9) / (r + 9)^2,
Multiply numerator and denomjnator by
(r + 9)^2 and get:
(r + 9)^-1 (r - 9) / 1,
Multiply numerator and denominator by
(r + 9)^1:
(r - 9) / (r + 9).
4. (v^2 - (16/25)v + 100) / (v - 4)/20
Multiply numerator and denominator by 25 and get:
(V^2 - 16V + 100) / 5(V - 4) / 4,
Multiply numerator and denominqator by
4/5 and get:
4/5(v^2 - 16v + 100) / (v - 4).
6. 2 / (z + 4) = 1 / (z - 6),
Cross multiply:
2z - 12 = z + 4,
2z - z = 4 + 12,
z = 16.
_4y_____ _ ___3y________
y^2+6y+5 y^2+5y+4
1. To subtract and simplify the expression 18/y - 8 - 18/(8 - y), we need to find a common denominator. The common denominator in this case is y(8 - y).
First, we multiply the numerator and denominator of the first fraction by (8 - y):
(18/y) * (8 - y)/(8 - y) = (18(8 - y))/(y(8 - y)) = (144 - 18y)/(y(8 - y))
Next, we multiply the numerator and denominator of the second fraction by y:
(18/(8 - y)) * y/y = (18y)/(y(8 - y))
Now that the denominators are the same, we can combine the numerators:
(144 - 18y)/(y(8 - y)) - (18y)/(y(8 - y)) = (144 - 18y - 18y)/(y(8 - y)) = (144 - 36y)/(y(8 - y))
This is the simplified result.
2. To subtract and simplify the expression (v - 5)/v - (7v - 31)/(11v), we need to find a common denominator. The common denominator in this case is 11v.
First, we multiply the numerator and denominator of the first fraction by 11:
[(v - 5)/v] * 11/11 = [(11v - 55)/(11v)]
Next, we multiply the numerator and denominator of the second fraction by v:
[(7v - 31)/(11v)] * v/v = [(7v^2 - 31v)/(11v^2)]
Now that the denominators are the same, we can combine the numerators:
[(11v - 55)/(11v)] - [(7v^2 - 31v)/(11v^2)] = [(11v^2 - 55v - 7v^2 + 31v)/(11v^2)] = [(4v^2 - 24v)/(11v^2)]
This is the simplified result.
3. To simplify the expression (r^2 - 81)/(r + 9)^2 by removing factors of 1, we factor the numerator and denominator.
The numerator is a difference of squares and can be factored as (r + 9)(r - 9).
The denominator is already factored as (r + 9)(r + 9) or (r + 9)^2.
Now, we can remove the common factor of (r + 9):
(r + 9)(r - 9)/(r + 9)(r + 9) = (r - 9)/(r + 9)
This is the simplified result.
4. To divide and simplify the expression (v^2 - 16)/(25v + 100) divided by (v - 4)/20, we can convert it to multiplication by taking the reciprocal of the second fraction.
The reciprocal of (v - 4)/20 is 20/(v - 4).
Now, we can rewrite the expression as:
[(v^2 - 16)/(25v + 100)] * [20/(v - 4)]
Next, factor the numerator as a difference of squares:
[(v + 4)(v - 4)/(25v + 100)] * [20/(v - 4)]
Now, we can cancel out the common factor of (v - 4):
[(v + 4)/25] * [20/1] = (v + 4)/5
This is the simplified result.
5. To find all rational numbers for which the rational expression -18/7w is undefined, we need to identify the values of w that would result in a denominator of zero.
In this case, the expression is undefined when the denominator 7w equals zero. So we need to solve the equation 7w = 0:
7w = 0
w = 0/7
w = 0
Therefore, the rational expression is undefined when w = 0.
6. To solve the equation 2/(z + 4) = 1/(z - 6), we can first cross-multiply to eliminate the denominators.
Cross-multiplying gives us:
2(z - 6) = 1(z + 4)
Next, distribute and simplify:
2z - 12 = z + 4
Now, we can isolate the variable z by subtracting z from both sides and adding 12 to both sides:
2z - z = 4 + 12
Simplifying:
z = 16
Therefore, the solution to the equation is z = 16.