Solve x/(x − 5) = 6
Rational equation, worked out line by line the way a teacher would write it.
Answer
| Solution | x = 6 |
Step-by-step solution
8 steps-
1 Given\frac{x}{x - 5} = 6
Solve for x.
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2 Note the values that make a denominator zero — they can never be solutions
Dividing by zero is undefined, so a value that makes a denominator 0 can never be a solution, even if it turns up as an answer later.
x \neq 5 -
3 Multiply every term by the least common denominator x - 5
Multiplying every term on both sides by the least common denominator x - 5 cancels every fraction at once. The equation stays balanced because all terms were multiplied by the same thing.
\frac{x}{x - 5} \left(x - 5\right) = \left(x - 5\right) 6This clears all the fractions.
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4 Cancel and simplifyx = 6 x - 30
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5 Subtract 6x from both sides
An equation stays true as long as you do the same thing to both sides, like adding the same weight to both pans of a balance. Adding or subtracting the same amount on both sides moves a term to the other side, where it appears with the opposite sign.
- 5 x = -30Collect every term containing x on the left.
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6 Divide both sides by -5
Dividing both sides by -5 undoes the multiplication by -5, so the variable is left on its own. Both sides change in the same way, so the equation stays true (dividing by 0 is the one thing that is never allowed).
x = \frac{-30}{-5} -
7 Simplifyx = 6
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8 Check\begin{aligned}x = 6:\quad 6 = 6\quad\checkmark\end{aligned}
Substituting each solution back makes both sides equal.
Check by substitution
| x | Left side | Right side | |
|---|---|---|---|
| 6 | 6 | 6 | ✓ |
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