Solve by Factoring 1/x+1/(x-7)=(x-6)/(x-7)

$\frac{1}{x} + \frac{1}{x - 7} = \frac{x - 6}{x - 7}$

Move $\frac{x - 6}{x - 7}$ to the left side of the equation by subtracting it from both sides.

$\frac{1}{x} + \frac{1}{x - 7} - \frac{x - 6}{x - 7} = 0$

Simplify $\frac{1}{x} + \frac{1}{x - 7} - \frac{x - 6}{x - 7}$ .

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To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{1}{x} \cdot \frac{x - 7}{x - 7} + \frac{1}{x - 7} - \frac{x - 6}{x - 7} = 0$

To write $\frac{1}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x} \cdot \frac{x - 7}{x - 7} + \frac{1}{x - 7} \cdot \frac{x}{x} - \frac{x - 6}{x - 7} = 0$

Write each expression with a common denominator of $x (x - 7)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{1}{x}$ and $\frac{x - 7}{x - 7}$ .

$\frac{x - 7}{x (x - 7)} + \frac{1}{x - 7} \cdot \frac{x}{x} - \frac{x - 6}{x - 7} = 0$

Multiply $\frac{1}{x - 7}$ and $\frac{x}{x}$ .

$\frac{x - 7}{x (x - 7)} + \frac{x}{(x - 7) x} - \frac{x - 6}{x - 7} = 0$

Reorder the factors of $(x - 7) x$ .

$\frac{x - 7}{x (x - 7)} + \frac{x}{x (x - 7)} - \frac{x - 6}{x - 7} = 0$

Combine the numerators over the common denominator.

$\frac{x - 7 + x}{x (x - 7)} - \frac{x - 6}{x - 7} = 0$

Add $x$ and $x$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{x - 6}{x - 7} = 0$

To write $- \frac{x - 6}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{x - 6}{x - 7} \cdot \frac{x}{x} = 0$

Write each expression with a common denominator of $x (x - 7)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{x - 6}{x - 7}$ and $\frac{x}{x}$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{(x - 6) x}{(x - 7) x} = 0$

Reorder the factors of $(x - 7) x$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{(x - 6) x}{x (x - 7)} = 0$

Combine the numerators over the common denominator.

$\frac{2 x - 7 - (x - 6) x}{x (x - 7)} = 0$

Simplify the numerator.

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Apply the distributive property.

$\frac{2 x - 7 + (- x - - 6) x}{x (x - 7)} = 0$

Multiply $- 1$ by $- 6$ .

$\frac{2 x - 7 + (- x + 6) x}{x (x - 7)} = 0$

Apply the distributive property.

$\frac{2 x - 7 - x \cdot x + 6 x}{x (x - 7)} = 0$

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

$\frac{2 x - 7 - (x \cdot x) + 6 x}{x (x - 7)} = 0$

Multiply $x$ by $x$ .

$\frac{2 x - 7 - x^{2} + 6 x}{x (x - 7)} = 0$

Add $2 x$ and $6 x$ .

$\frac{8 x - 7 - x^{2}}{x (x - 7)} = 0$

Reorder terms.

$\frac{- x^{2} + 8 x - 7}{x (x - 7)} = 0$

Factor by grouping.

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For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 7 = 7$ and whose sum is $b = 8$ .

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Factor $8$ out of $8 x$ .

$\frac{- x^{2} + 8 (x) - 7}{x (x - 7)} = 0$

Rewrite $8$ as $1$ plus $7$

$\frac{- x^{2} + (1 + 7) x - 7}{x (x - 7)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 1 x + 7 x - 7}{x (x - 7)} = 0$

Multiply $x$ by $1$ .

$\frac{- x^{2} + x + 7 x - 7}{x (x - 7)} = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$\frac{(- x^{2} + x) + 7 x - 7}{x (x - 7)} = 0$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x + 1) - 7 (- x + 1)}{x (x - 7)} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$\frac{(- x + 1) (x - 7)}{x (x - 7)} = 0$

Simplify terms.

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Cancel the common factor of $x - 7$ .

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Cancel the common factor.

$\frac{(- x + 1) (x - 7)}{x (x - 7)} = 0$

Rewrite the expression.

$\frac{- x + 1}{x} = 0$

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 1}{x} = 0$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (x) - 1 (- 1)}{x} = 0$

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$\frac{- (x - 1)}{x} = 0$

Simplify the expression.

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Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$\frac{- 1 (x - 1)}{x} = 0$

Move the negative in front of the fraction.

$- \frac{x - 1}{x} = 0$

To write $\frac{1}{x}$ as a fraction with a common denominator, multiply by $\frac{x - 7}{x - 7}$ .

$\frac{1}{x} \cdot \frac{x - 7}{x - 7} + \frac{1}{x - 7} - \frac{x - 6}{x - 7} = 0$

To write $\frac{1}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{1}{x} \cdot \frac{x - 7}{x - 7} + \frac{1}{x - 7} \cdot \frac{x}{x} - \frac{x - 6}{x - 7} = 0$

Write each expression with a common denominator of $x (x - 7)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{1}{x}$ and $\frac{x - 7}{x - 7}$ .

$\frac{x - 7}{x (x - 7)} + \frac{1}{x - 7} \cdot \frac{x}{x} - \frac{x - 6}{x - 7} = 0$

Multiply $\frac{1}{x - 7}$ and $\frac{x}{x}$ .

$\frac{x - 7}{x (x - 7)} + \frac{x}{(x - 7) x} - \frac{x - 6}{x - 7} = 0$

Reorder the factors of $(x - 7) x$ .

$\frac{x - 7}{x (x - 7)} + \frac{x}{x (x - 7)} - \frac{x - 6}{x - 7} = 0$

Combine the numerators over the common denominator.

$\frac{x - 7 + x}{x (x - 7)} - \frac{x - 6}{x - 7} = 0$

Add $x$ and $x$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{x - 6}{x - 7} = 0$

To write $- \frac{x - 6}{x - 7}$ as a fraction with a common denominator, multiply by $\frac{x}{x}$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{x - 6}{x - 7} \cdot \frac{x}{x} = 0$

Write each expression with a common denominator of $x (x - 7)$ , by multiplying each by an appropriate factor of $1$ .

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Multiply $\frac{x - 6}{x - 7}$ and $\frac{x}{x}$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{(x - 6) x}{(x - 7) x} = 0$

Reorder the factors of $(x - 7) x$ .

$\frac{2 x - 7}{x (x - 7)} - \frac{(x - 6) x}{x (x - 7)} = 0$

Combine the numerators over the common denominator.

$\frac{2 x - 7 - (x - 6) x}{x (x - 7)} = 0$

Simplify the numerator.

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Apply the distributive property.

$\frac{2 x - 7 + (- x + 6) x}{x (x - 7)} = 0$

Multiply $- 1$ by $- 6$ .

$\frac{2 x - 7 + (- x + 6) x}{x (x - 7)} = 0$

Apply the distributive property.

$\frac{2 x - 7 - x \cdot x + 6 x}{x (x - 7)} = 0$

Multiply $x$ by $x$ by adding the exponents.

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Move $x$ .

$\frac{2 x - 7 - (x \cdot x) + 6 x}{x (x - 7)} = 0$

Multiply $x$ by $x$ .

$\frac{2 x - 7 - x^{2} + 6 x}{x (x - 7)} = 0$

Add $2 x$ and $6 x$ .

$\frac{8 x - 7 - x^{2}}{x (x - 7)} = 0$

Reorder terms.

$\frac{- x^{2} + 8 x - 7}{x (x - 7)} = 0$

Factor by grouping.

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For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = - 1 \cdot - 7 = 7$ and whose sum is $b = 8$ .

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Factor $8$ out of $8 x$ .

$\frac{- x^{2} + 8 (x) - 7}{x (x - 7)} = 0$

Rewrite $8$ as $1$ plus $7$

$\frac{- x^{2} + (1 + 7) x - 7}{x (x - 7)} = 0$

Apply the distributive property.

$\frac{- x^{2} + 1 x + 7 x - 7}{x (x - 7)} = 0$

Multiply $x$ by $1$ .

$\frac{- x^{2} + x + 7 x - 7}{x (x - 7)} = 0$

Factor out the greatest common factor from each group.

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Group the first two terms and the last two terms.

$\frac{(- x^{2} + x) + 7 x - 7}{x (x - 7)} = 0$

Factor out the greatest common factor (GCF) from each group.

$\frac{x (- x + 1) - 7 (- x + 1)}{x (x - 7)} = 0$

Factor the polynomial by factoring out the greatest common factor, $- x + 1$ .

$\frac{(- x + 1) (x - 7)}{x (x - 7)} = 0$

Simplify terms.

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Cancel the common factor of $x - 7$ .

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Cancel the common factor.

$\frac{(- x + 1) (x - 7)}{x (x - 7)} = 0$

Rewrite the expression.

$\frac{- x + 1}{x} = 0$

Factor $- 1$ out of $- x$ .

$\frac{- (x) + 1}{x} = 0$

Rewrite $1$ as $- 1 (- 1)$ .

$\frac{- (x) - 1 \cdot - 1}{x} = 0$

Factor $- 1$ out of $- (x) - 1 (- 1)$ .

$\frac{- (x - 1)}{x} = 0$

Simplify the expression.

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Rewrite $- (x - 1)$ as $- 1 (x - 1)$ .

$\frac{- 1 (x - 1)}{x} = 0$

Move the negative in front of the fraction.

$- \frac{x - 1}{x} = 0$

Find the LCD of the terms in the equation.

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Finding the LCD of a list of values is the same as finding the LCM of the denominators of those values.

$x, 1$

Since $x, 1$ contain both numbers and variables, there are two steps to find the LCM. Find LCM for the numeric part $1, 1$ then find LCM for the variable part $x^{1}$ .

The LCM is the smallest positive number that all of the numbers divide into evenly.

1. List the prime factors of each number.

2. Multiply each factor the greatest number of times it occurs in either number.

The number $1$ is not a prime number because it only has one positive factor, which is itself.

Not prime

The LCM of $1, 1$ is the result of multiplying all prime factors the greatest number of times they occur in either number.

$1$

The factor for $x^{1}$ is $x$ itself.

$x^{1} = x$

$x$ occurs $1$ time.

The LCM of $x^{1}$ is the result of multiplying all prime factors the greatest number of times they occur in either term.

$x$

Multiply each term by $x$ and simplify.

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Multiply each term in $- \frac{x - 1}{x} = 0$ by $x$ in order to remove all the denominators from the equation.

$- \frac{x - 1}{x} \cdot x = 0 \cdot x$

Simplify $- \frac{x - 1}{x} \cdot x$ .

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Cancel the common factor of $x$ .

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Move the leading negative in $- \frac{x - 1}{x}$ into the numerator.

$\frac{- (x - 1)}{x} \cdot x = 0 \cdot x$

Cancel the common factor.

$\frac{- (x - 1)}{x} \cdot x = 0 \cdot x$

Rewrite the expression.

$- (x - 1) = 0 \cdot x$

Apply the distributive property.

$- x - - 1 = 0 \cdot x$

Multiply $- 1$ by $- 1$ .

$- x + 1 = 0 \cdot x$

Multiply $0$ by $x$ .

$- x + 1 = 0$

Solve the equation.

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Subtract $1$ from both sides of the equation.

$- x = - 1$

Multiply each term in $- x = - 1$ by $- 1$

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Multiply each term in $- x = - 1$ by $- 1$ .

$(- x) \cdot - 1 = (- 1) \cdot - 1$

Multiply $(- x) \cdot - 1$ .

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Multiply $- 1$ by $- 1$ .

$1 x = (- 1) \cdot - 1$

Multiply $x$ by $1$ .

$x = (- 1) \cdot - 1$

Multiply $- 1$ by $- 1$ .

$x = 1$

Solve by Factoring 1/x+1/(x-7)=(x-6)/(x-7)

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