I assume this is a proportion - one fraction that equals another fraction. Here is the equation with parentheses to help avoid confusion.
(z + 4) / (z + 5) = -1 / (z + 5)
Use the Means Extremes Property. According to the Means Extremes Property...
If a/b = c/d, the ad = bc.
(z + 4)(z + 5) = -1(z + 5)
z^2 + 9z + 20 = -1z + -5
z^2 + 10z + 25 = 0
Factor the trinomial.
(z + 5)^2 = 0
z + 5 = 0
z = -5
However, z `!=` -5 because that would cause the denominators of the fractions to equal 0, which is impossible.
If this problem is not a proportion, the problem would be solved differently.
z + 4/z + 5 = -1/z + 5
Multiply both sides by z to eliminate the fractions.
z^2 + 4 + 5z = -1 + 5z
z^2 + 4 = -1
z^2 = -5
z = sqrt(-5)
You cannot square root a negative number, therefore there is no real solution.
Either way, the equation has no solution.
Is it possible you mistyped the equation, or forgot to insert parentheses?
We have to solve (z+4)/(z+5) = -1/(z+5) for z
(z+4)/(z+5) = -1/(z+5)
=> z + 4 = -1
=> z = -5
but at z = -5, (z + 5) = 0
As the denominator cannot be equal to zero, z cannot equal -5.
The equation does not have a solution.
This is the equation
multiply z+5 to both sides, adding a restriction z could not be -5
the restriction says that z could not be -5
the answer is DNE (does not exist)
another way of doing this is to move the -1/(z+5) to the left side
This is NEVER true, so the answer is DNE
The solution to this equation is DNE(Does not exist)