Algebra Examples

$\frac{x}{10} = \frac{10}{25 - x}$

Step 1

Multiply the numerator of the first fraction by the denominator of the second fraction. Set this equal to the product of the denominator of the first fraction and the numerator of the second fraction.

$x (25 - x) = 10 \cdot 10$

Step 2

Solve the equation for $x$ .

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Step 2.1

Simplify $x (25 - x)$ .

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Step 2.1.1

Simplify by multiplying through.

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Step 2.1.1.1

Apply the distributive property.

$x \cdot 25 + x (- x) = 10 \cdot 10$

Step 2.1.1.2

Reorder.

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Step 2.1.1.2.1

Move $25$ to the left of $x$ .

$25 \cdot x + x (- x) = 10 \cdot 10$

Step 2.1.1.2.2

Rewrite using the commutative property of multiplication.

$25 \cdot x - x \cdot x = 10 \cdot 10$

Step 2.1.2

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

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Step 2.1.2.1

Move $x$ .

$25 x - (x \cdot x) = 10 \cdot 10$

Step 2.1.2.2

Multiply $x$ by $x$ .

$25 x - x^{2} = 10 \cdot 10$

Step 2.2

Multiply $10$ by $10$ .

$25 x - x^{2} = 100$

Step 2.3

Subtract $100$ from both sides of the equation.

$25 x - x^{2} - 100 = 0$

Step 2.4

Factor the left side of the equation.

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Step 2.4.1

Factor $- 1$ out of $25 x - x^{2} - 100$ .

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Step 2.4.1.1

Reorder $25 x$ and $- x^{2}$ .

$- x^{2} + 25 x - 100 = 0$

Step 2.4.1.2

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) + 25 x - 100 = 0$

Step 2.4.1.3

Factor $- 1$ out of $25 x$ .

$- (x^{2}) - (- 25 x) - 100 = 0$

Step 2.4.1.4

Rewrite $- 100$ as $- 1 (100)$ .

$- (x^{2}) - (- 25 x) - 1 \cdot 100 = 0$

Step 2.4.1.5

Factor $- 1$ out of $- (x^{2}) - (- 25 x)$ .

$- (x^{2} - 25 x) - 1 \cdot 100 = 0$

Step 2.4.1.6

Factor $- 1$ out of $- (x^{2} - 25 x) - 1 (100)$ .

$- (x^{2} - 25 x + 100) = 0$

Step 2.4.2

Factor.

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Step 2.4.2.1

Factor $x^{2} - 25 x + 100$ using the AC method.

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Step 2.4.2.1.1

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 $100$ and whose sum is $- 25$ .

$- 20, - 5$

Step 2.4.2.1.2

Write the factored form using these integers.

$- ((x - 20) (x - 5)) = 0$

Step 2.4.2.2

Remove unnecessary parentheses.

$- (x - 20) (x - 5) = 0$

Step 2.5

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

$x - 20 = 0$

$x - 5 = 0$

Step 2.6

Set $x - 20$ equal to $0$ and solve for $x$ .

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Step 2.6.1

Set $x - 20$ equal to $0$ .

$x - 20 = 0$

Step 2.6.2

Add $20$ to both sides of the equation.

$x = 20$

Step 2.7

Set $x - 5$ equal to $0$ and solve for $x$ .

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Step 2.7.1

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Step 2.7.2

Add $5$ to both sides of the equation.

$x = 5$

Step 2.8

The final solution is all the values that make $- (x - 20) (x - 5) = 0$ true.

$x = 20, 5$