Solve for x 7+(3x+7)/(x+8)=x

$7 + \frac{3 x + 7}{x + 8} = x$

Step 1

Find the LCD of the terms in the equation.

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

$1, x + 8, 1$

Remove parentheses.

$1, x + 8, 1$

The LCM of one and any expression is the expression.

$x + 8$

Step 2

Multiply each term in $7 + \frac{3 x + 7}{x + 8} = x$ by $x + 8$ to eliminate the fractions.

Multiply each term in $7 + \frac{3 x + 7}{x + 8} = x$ by $x + 8$ .

$7 (x + 8) + \frac{3 x + 7}{x + 8} (x + 8) = x (x + 8)$

Simplify the left side.

Simplify each term.

Apply the distributive property.

$7 x + 7 \cdot 8 + \frac{3 x + 7}{x + 8} (x + 8) = x (x + 8)$

Multiply $7$ by $8$ .

$7 x + 56 + \frac{3 x + 7}{x + 8} (x + 8) = x (x + 8)$

Cancel the common factor of $x + 8$ .

Cancel the common factor.

$7 x + 56 + \frac{3 x + 7}{x + 8} (x + 8) = x (x + 8)$

Rewrite the expression.

$7 x + 56 + 3 x + 7 = x (x + 8)$

Simplify by adding terms.

Add $7 x$ and $3 x$ .

$10 x + 56 + 7 = x (x + 8)$

Add $56$ and $7$ .

$10 x + 63 = x (x + 8)$

Simplify the right side.

Apply the distributive property.

$10 x + 63 = x \cdot x + x \cdot 8$

Simplify the expression.

Multiply $x$ by $x$ .

$10 x + 63 = x^{2} + x \cdot 8$

Move $8$ to the left of $x$ .

$10 x + 63 = x^{2} + 8 x$

Step 3

Solve the equation.

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 8 x = 10 x + 63$

Move all terms containing $x$ to the left side of the equation.

Subtract $10 x$ from both sides of the equation.

$x^{2} + 8 x - 10 x = 63$

Subtract $10 x$ from $8 x$ .

$x^{2} - 2 x = 63$

Subtract $63$ from both sides of the equation.

$x^{2} - 2 x - 63 = 0$

Factor $x^{2} - 2 x - 63$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 63$ and whose sum is $- 2$ .

$- 9, 7$

Write the factored form using these integers.

$(x - 9) (x + 7) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x - 9 = 0$

$x + 7 = 0$

Set $x - 9$ equal to $0$ and solve for $x$ .

Set $x - 9$ equal to $0$ .

$x - 9 = 0$

Add $9$ to both sides of the equation.

$x = 9$

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

Subtract $7$ from both sides of the equation.

$x = - 7$

The final solution is all the values that make $(x - 9) (x + 7) = 0$ true.

$x = 9, - 7$

Solve for x 7+(3x+7)/(x+8)=x

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