help me solve
6/z+8 = 5/z-3?
do i first find the lcd?
I will assume that you mean
6/(z+8) = 5/(z-3)
Parentheses are very important when you write equations on a single line, as here.
Cross multiply to get
6z - 18 = 5z + 40
Now combine terms
z = 58
You could look for an LCM, but that step is not necessary. The answer would be the same.
Maybe I wrote the problem wrong...
6 = 5
z+8 4+x
is that the same as what i put earlier?
Yes, finding the least common denominator (LCD) is the first step in solving this equation.
To find the LCD, you need to factor the denominators and find the common factors. In this case, the denominators are "z" and "z-3".
The factors of z are (z) and (1). The factors of z-3 are (z-3) and (1).
Since there are no common factors other than 1, the least common denominator is simply z(z-3).
Now, rewrite the equation with the LCD:
(6/z)(z(z-3)) + 8(z(z-3)) = (5/z)(z(z-3)) - 3(z(z-3))
Simplify the equation:
6z(z-3) + 8z(z-3) = 5z(z-3) - 3z(z-3)
Expand the expressions:
6z^2 - 18z + 8z^2 - 24z = 5z^2 - 15z - 3z^2 + 9z
Combine like terms:
14z^2 - 42z = 2z^2 - 9z
Move all the terms to one side:
14z^2 - 42z - 2z^2 + 9z = 0
Combine like terms again:
12z^2 - 33z = 0
Now you can solve for z by factoring or using the quadratic formula. Factor out common terms:
z(12z - 33) = 0
Now set each factor equal to zero:
z = 0
and
12z - 33 = 0
Solve for z:
12z = 33
z = 33/12
z = 11/4
So, the values of z that satisfy the equation are z = 0 and z = 11/4.