Algebra Examples

$\frac{t}{t - 6} = \frac{t + 6}{- 9}$

Step 1

Subtract $\frac{t + 6}{- 9}$ from both sides of the equation.

$\frac{t}{t - 6} - \frac{t + 6}{- 9} = 0$

Step 2

Simplify $\frac{t}{t - 6} - \frac{t + 6}{- 9}$ .

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Step 2.1

Simplify each term.

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Step 2.1.1

Move the negative in front of the fraction.

$\frac{t}{t - 6} - - \frac{t + 6}{9} = 0$

Step 2.1.2

Multiply $- - \frac{t + 6}{9}$ .

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Step 2.1.2.1

Multiply $- 1$ by $- 1$ .

$\frac{t}{t - 6} + 1 \frac{t + 6}{9} = 0$

Step 2.1.2.2

Multiply $\frac{t + 6}{9}$ by $1$ .

$\frac{t}{t - 6} + \frac{t + 6}{9} = 0$

Step 2.2

To write $\frac{t}{t - 6}$ as a fraction with a common denominator, multiply by $\frac{9}{9}$ .

$\frac{t}{t - 6} \cdot \frac{9}{9} + \frac{t + 6}{9} = 0$

Step 2.3

To write $\frac{t + 6}{9}$ as a fraction with a common denominator, multiply by $\frac{t - 6}{t - 6}$ .

$\frac{t}{t - 6} \cdot \frac{9}{9} + \frac{t + 6}{9} \cdot \frac{t - 6}{t - 6} = 0$

Step 2.4

Write each expression with a common denominator of $(t - 6) \cdot 9$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.4.1

Multiply $\frac{t}{t - 6}$ by $\frac{9}{9}$ .

$\frac{t \cdot 9}{(t - 6) \cdot 9} + \frac{t + 6}{9} \cdot \frac{t - 6}{t - 6} = 0$

Step 2.4.2

Multiply $\frac{t + 6}{9}$ by $\frac{t - 6}{t - 6}$ .

$\frac{t \cdot 9}{(t - 6) \cdot 9} + \frac{(t + 6) (t - 6)}{9 (t - 6)} = 0$

Step 2.4.3

Reorder the factors of $(t - 6) \cdot 9$ .

$\frac{t \cdot 9}{9 (t - 6)} + \frac{(t + 6) (t - 6)}{9 (t - 6)} = 0$

Step 2.5

Combine the numerators over the common denominator.

$\frac{t \cdot 9 + (t + 6) (t - 6)}{9 (t - 6)} = 0$

Step 2.6

Simplify the numerator.

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Step 2.6.1

Move $9$ to the left of $t$ .

$\frac{9 \cdot t + (t + 6) (t - 6)}{9 (t - 6)} = 0$

Step 2.6.2

Expand $(t + 6) (t - 6)$ using the FOIL Method.

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Step 2.6.2.1

Apply the distributive property.

$\frac{9 t + t (t - 6) + 6 (t - 6)}{9 (t - 6)} = 0$

Step 2.6.2.2

Apply the distributive property.

$\frac{9 t + t \cdot t + t \cdot - 6 + 6 (t - 6)}{9 (t - 6)} = 0$

Step 2.6.2.3

Apply the distributive property.

$\frac{9 t + t \cdot t + t \cdot - 6 + 6 t + 6 \cdot - 6}{9 (t - 6)} = 0$

Step 2.6.3

Combine the opposite terms in $t \cdot t + t \cdot - 6 + 6 t + 6 \cdot - 6$ .

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Step 2.6.3.1

Reorder the factors in the terms $t \cdot - 6$ and $6 t$ .

$\frac{9 t + t \cdot t - 6 t + 6 t + 6 \cdot - 6}{9 (t - 6)} = 0$

Step 2.6.3.2

Add $- 6 t$ and $6 t$ .

$\frac{9 t + t \cdot t + 0 + 6 \cdot - 6}{9 (t - 6)} = 0$

Step 2.6.3.3

Add $t \cdot t$ and $0$ .

$\frac{9 t + t \cdot t + 6 \cdot - 6}{9 (t - 6)} = 0$

Step 2.6.4

Simplify each term.

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Step 2.6.4.1

Multiply $t$ by $t$ .

$\frac{9 t + t^{2} + 6 \cdot - 6}{9 (t - 6)} = 0$

Step 2.6.4.2

Multiply $6$ by $- 6$ .

$\frac{9 t + t^{2} - 36}{9 (t - 6)} = 0$

Step 2.6.5

Reorder terms.

$\frac{t^{2} + 9 t - 36}{9 (t - 6)} = 0$

Step 2.6.6

Factor $t^{2} + 9 t - 36$ using the AC method.

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Step 2.6.6.1

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 $- 36$ and whose sum is $9$ .

$- 3, 12$

Step 2.6.6.2

Write the factored form using these integers.

$\frac{(t - 3) (t + 12)}{9 (t - 6)} = 0$

Step 3

Set the numerator equal to zero.

$(t - 3) (t + 12) = 0$

Step 4

Solve the equation for $t$ .

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Step 4.1

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

$t - 3 = 0$

$t + 12 = 0$

Step 4.2

Set $t - 3$ equal to $0$ and solve for $t$ .

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Step 4.2.1

Set $t - 3$ equal to $0$ .

$t - 3 = 0$

Step 4.2.2

Add $3$ to both sides of the equation.

$t = 3$

Step 4.3

Set $t + 12$ equal to $0$ and solve for $t$ .

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Step 4.3.1

Set $t + 12$ equal to $0$ .

$t + 12 = 0$

Step 4.3.2

Subtract $12$ from both sides of the equation.

$t = - 12$

Step 4.4

The final solution is all the values that make $(t - 3) (t + 12) = 0$ true.

$t = 3, - 12$