Finite Math Examples

$(3 a - 2) = \frac{a + 25}{a + 1}$

Step 1

Move all terms not containing $a$ to the right side of the equation.

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Step 1.1

Add $2$ to both sides of the equation.

$3 a = \frac{a + 25}{a + 1} + 2$

Step 1.2

Split the fraction $\frac{a + 25}{a + 1}$ into two fractions.

$3 a = \frac{a}{a + 1} + \frac{25}{a + 1} + 2$

Step 1.3

Find the common denominator.

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Step 1.3.1

Write $2$ as a fraction with denominator $1$ .

$3 a = \frac{a}{a + 1} + \frac{25}{a + 1} + \frac{2}{1}$

Step 1.3.2

Multiply $\frac{2}{1}$ by $\frac{a + 1}{a + 1}$ .

$3 a = \frac{a}{a + 1} + \frac{25}{a + 1} + \frac{2}{1} \cdot \frac{a + 1}{a + 1}$

Step 1.3.3

Multiply $\frac{2}{1}$ by $\frac{a + 1}{a + 1}$ .

$3 a = \frac{a}{a + 1} + \frac{25}{a + 1} + \frac{2 (a + 1)}{a + 1}$

Step 1.4

Combine the numerators over the common denominator.

$3 a = \frac{a + 25 + 2 (a + 1)}{a + 1}$

Step 1.5

Simplify each term.

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Step 1.5.1

Apply the distributive property.

$3 a = \frac{a + 25 + 2 a + 2 \cdot 1}{a + 1}$

Step 1.5.2

Multiply $2$ by $1$ .

$3 a = \frac{a + 25 + 2 a + 2}{a + 1}$

Step 1.6

Add $a$ and $2 a$ .

$3 a = \frac{3 a + 25 + 2}{a + 1}$

Step 1.7

Add $25$ and $2$ .

$3 a = \frac{3 a + 27}{a + 1}$

Step 1.8

Factor $3$ out of $3 a + 27$ .

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Step 1.8.1

Factor $3$ out of $3 a$ .

$3 a = \frac{3 (a) + 27}{a + 1}$

Step 1.8.2

Factor $3$ out of $27$ .

$3 a = \frac{3 a + 3 \cdot 9}{a + 1}$

Step 1.8.3

Factor $3$ out of $3 a + 3 \cdot 9$ .

$3 a = \frac{3 (a + 9)}{a + 1}$

Step 2

Find the LCD of the terms in the equation.

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Step 2.1

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

$1, a + 1$

Step 2.2

Remove parentheses.

$1, a + 1$

Step 2.3

The LCM of one and any expression is the expression.

$a + 1$

Step 3

Multiply each term in $3 a = \frac{3 (a + 9)}{a + 1}$ by $a + 1$ to eliminate the fractions.

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Step 3.1

Multiply each term in $3 a = \frac{3 (a + 9)}{a + 1}$ by $a + 1$ .

$3 a (a + 1) = \frac{3 (a + 9)}{a + 1} (a + 1)$

Step 3.2

Simplify the left side.

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Step 3.2.1

Apply the distributive property.

$3 a \cdot a + 3 a \cdot 1 = \frac{3 (a + 9)}{a + 1} (a + 1)$

Step 3.2.2

Multiply $a$ by $a$ by adding the exponents.

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Step 3.2.2.1

Move $a$ .

$3 (a \cdot a) + 3 a \cdot 1 = \frac{3 (a + 9)}{a + 1} (a + 1)$

Step 3.2.2.2

Multiply $a$ by $a$ .

$3 a^{2} + 3 a \cdot 1 = \frac{3 (a + 9)}{a + 1} (a + 1)$

Step 3.2.3

Multiply $3$ by $1$ .

$3 a^{2} + 3 a = \frac{3 (a + 9)}{a + 1} (a + 1)$

Step 3.3

Simplify the right side.

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Step 3.3.1

Cancel the common factor of $a + 1$ .

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Step 3.3.1.1

Cancel the common factor.

$3 a^{2} + 3 a = \frac{3 (a + 9)}{a + 1} (a + 1)$

Step 3.3.1.2

Rewrite the expression.

$3 a^{2} + 3 a = 3 (a + 9)$

Step 3.3.2

Apply the distributive property.

$3 a^{2} + 3 a = 3 a + 3 \cdot 9$

Step 3.3.3

Multiply $3$ by $9$ .

$3 a^{2} + 3 a = 3 a + 27$

Step 4

Solve the equation.

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Step 4.1

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

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Step 4.1.1

Subtract $3 a$ from both sides of the equation.

$3 a^{2} + 3 a - 3 a = 27$

Step 4.1.2

Combine the opposite terms in $3 a^{2} + 3 a - 3 a$ .

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Step 4.1.2.1

Subtract $3 a$ from $3 a$ .

$3 a^{2} + 0 = 27$

Step 4.1.2.2

Add $3 a^{2}$ and $0$ .

$3 a^{2} = 27$

Step 4.2

Divide each term in $3 a^{2} = 27$ by $3$ and simplify.

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Step 4.2.1

Divide each term in $3 a^{2} = 27$ by $3$ .

$\frac{3 a^{2}}{3} = \frac{27}{3}$

Step 4.2.2

Simplify the left side.

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Step 4.2.2.1

Cancel the common factor of $3$ .

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Step 4.2.2.1.1

Cancel the common factor.

$\frac{3 a^{2}}{3} = \frac{27}{3}$

Step 4.2.2.1.2

Divide $a^{2}$ by $1$ .

$a^{2} = \frac{27}{3}$

Step 4.2.3

Simplify the right side.

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Step 4.2.3.1

Divide $27$ by $3$ .

$a^{2} = 9$

Step 4.3

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$a = \pm \sqrt{9}$

Step 4.4

Simplify $\pm \sqrt{9}$ .

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Step 4.4.1

Rewrite $9$ as $3^{2}$ .

$a = \pm \sqrt{3^{2}}$

Step 4.4.2

Pull terms out from under the radical, assuming positive real numbers.

$a = \pm 3$

Step 4.5

The complete solution is the result of both the positive and negative portions of the solution.

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Step 4.5.1

First, use the positive value of the $\pm$ to find the first solution.

$a = 3$

Step 4.5.2

Next, use the negative value of the $\pm$ to find the second solution.

$a = - 3$

Step 4.5.3

The complete solution is the result of both the positive and negative portions of the solution.

$a = 3, - 3$