Simplify the sum. State any restrictions on the variable. Show your work.
2z+7/z-5 + 2z+4/z-5
To simplify the sum, we can combine like terms:
(2z + 7)/(z - 5) + (2z + 4)/(z - 5)
Both fractions have the same denominator, so we can combine them:
(2z + 7 + 2z + 4)/(z - 5)
Combining like terms in the numerator:
4z + 11 / (z - 5)
Therefore, the simplified sum is 4z + 11 / (z - 5).
Restrictions: The variable z cannot equal 5, as this would result in a zero denominator.
To simplify the sum, we can combine the two fractions by finding a common denominator.
The common denominator for the two fractions is (z - 5).
Now let's rewrite the fractions with the common denominator:
2z + 7 / (z - 5) + 2z + 4 / (z - 5)
Next, we can combine the numerators:
(2z + 7 + 2z + 4) / (z - 5)
Simplifying the numerator:
4z + 11 / (z - 5)
Therefore, the simplified sum is (4z + 11) / (z - 5).
Restrictions on the variable:
The variable z cannot be equal to 5, as this would make the denominator zero, resulting in division by zero, which is undefined.
To simplify the sum (2z+7)/(z-5) + (2z+4)/(z-5), we can combine the fractions by finding a common denominator.
Step 1: Find the common denominator.
In this case, the common denominator is (z-5) since it appears in both fractions.
Step 2: Rewrite the fractions with the common denominator.
For the first fraction, (2z+7)/(z-5), we multiply the numerator and denominator by (z-5) to obtain:
(2z+7)*(z-5)/(z-5)*(z-5) = (2z^2 - 3z - 35)/(z-5)
For the second fraction, (2z+4)/(z-5), we multiply the numerator and denominator by (z-5) to obtain:
(2z+4)*(z-5)/(z-5)*(z-5) = (2z^2 - 6z - 20)/(z-5)
Step 3: Combine the fractions.
Now that both fractions have the same denominator, we can add them:
(2z^2 - 3z - 35)/(z-5) + (2z^2 - 6z - 20)/(z-5)
To combine them, we add the numerators together but keep the denominator the same:
(2z^2 - 3z - 35 + 2z^2 - 6z - 20)/(z-5)
Simplifying the numerator gives:
(4z^2 - 9z - 55)/(z-5)
Step 4: Check for restrictions on the variable.
To check for restrictions on the variable, we need to ensure that the denominator (z-5) does not equal zero. If it does, it would result in a division by zero error.
So, in this case, the restriction is z ≠ 5.
Therefore, simplifying the sum (2z+7)/(z-5) + (2z+4)/(z-5) gives (4z^2 - 9z - 55)/(z-5), with the restriction z ≠ 5.