To solve the equation [tex]\( \frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)} \)[/tex], let's proceed step by step:
1. Identify the common denominator:
The common denominator of the fractions on both sides of the equation is [tex]\( c(c-3) \)[/tex].
2. Express each term with the common denominator:
[tex]\[
\frac{1}{c-3} = \frac{c}{c(c-3)}, \quad \frac{1}{c} = \frac{c-3}{c(c-3)}
\][/tex]
3. Rewriting the equation with a common denominator:
[tex]\[
\frac{c}{c(c-3)} - \frac{c-3}{c(c-3)} = \frac{3}{c(c-3)}
\][/tex]
4. Combine the fractions on the left side:
[tex]\[
\frac{c - (c-3)}{c(c-3)} = \frac{3}{c(c-3)}
\][/tex]
5. Simplify the numerator on the left:
[tex]\[
\frac{c - c + 3}{c(c-3)} = \frac{3}{c(c-3)}
\][/tex]
[tex]\[
\frac{3}{c(c-3)} = \frac{3}{c(c-3)}
\][/tex]
6. Since both sides of the equation are equal:
This identity confirms that the equation holds true generally for [tex]\( c \)[/tex] except where the denominators are undefined, i.e., where [tex]\( c \)[/tex] causes division by zero.
7. Determine the values that make the denominators zero:
[tex]\[
c(c-3) \neq 0
\][/tex]
[tex]\[
c \neq 0 \quad \text{and} \quad c \neq 3
\][/tex]
8. Conclude that the equation has no solutions at [tex]\( c=0 \)[/tex] and [tex]\( c=3 \)[/tex]:
Since the integers [tex]\( c = 0 \)[/tex] and [tex]\( c = 3 \)[/tex] would cause division by zero, these values are not valid solutions.
Therefore, the solution to the equation [tex]\( \frac{1}{c-3} - \frac{1}{c} = \frac{3}{c(c-3)} \)[/tex] is:
[tex]\[
\text{No solution}
\][/tex]
