To solve the equation [tex]\(\frac{1}{x-9} + 3 = \frac{x}{x-9}\)[/tex], we need to follow several steps to isolate [tex]\(x\)[/tex].
Step 1: Equalize the fractions
Since both terms on the left-hand side of the equation have the same denominator [tex]\((x-9)\)[/tex], we can combine them into a single fraction:
[tex]\[
\frac{1}{x-9} + 3 = \frac{1 + 3(x-9)}{x-9}
\][/tex]
Step 2: Simplify the numerator
Next, simplify the expression in the numerator of the combined fraction:
[tex]\[
1 + 3(x-9) = 1 + 3x - 27 = 3x - 26
\][/tex]
Step 3: Substitute the simplified expression back into the fraction:
[tex]\[
\frac{3x - 26}{x-9}
\][/tex]
Step 4: Form the new equation with the right-hand side:
Now the equation looks like this:
[tex]\[
\frac{3x - 26}{x-9} = \frac{x}{x-9}
\][/tex]
Step 5: Since the denominators on both sides of the equation are the same, equate the numerators:
[tex]\[
3x - 26 = x
\][/tex]
Step 6: Solve for [tex]\(x\)[/tex]:
[tex]\[
3x - x = 26
\][/tex]
[tex]\[
2x = 26
\][/tex]
[tex]\[
x = 13
\][/tex]
Therefore, the solution to the equation is:
[tex]\[
x = 13
\][/tex]
