Algebra Examples

$\frac{14}{z} = 9 - z$

Step 1

Move all the expressions to the left side of the equation.

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Step 1.1

Subtract $9$ from both sides of the equation.

$\frac{14}{z} - 9 = - z$

Step 1.2

Add $z$ to both sides of the equation.

$\frac{14}{z} - 9 + z = 0$

Step 2

Reorder terms.

$z + \frac{14}{z} - 9 = 0$

Step 3

To write $z$ as a fraction with a common denominator, multiply by $\frac{z}{z}$ .

$\frac{z \cdot z}{z} + \frac{14}{z} - 9 = 0$

Step 4

Combine the numerators over the common denominator.

$\frac{z \cdot z + 14}{z} - 9 = 0$

Step 5

Multiply $z$ by $z$ .

$\frac{z^{2} + 14}{z} - 9 = 0$

Step 6

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

$\frac{z^{2} + 14}{z} - 9 \cdot \frac{z}{z} = 0$

Step 7

Combine $- 9$ and $\frac{z}{z}$ .

$\frac{z^{2} + 14}{z} + \frac{- 9 z}{z} = 0$

Step 8

Combine the numerators over the common denominator.

$\frac{z^{2} + 14 - 9 z}{z} = 0$

Step 9

Factor $z^{2} + 14 - 9 z$ using the AC method.

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Step 9.1

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 $14$ and whose sum is $- 9$ .

$- 7, - 2$

Step 9.2

Write the factored form using these integers.

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

Step 10

Set the numerator equal to zero.

$(z - 7) (z - 2) = 0$

Step 11

Solve the equation for $z$ .

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Step 11.1

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

$z - 7 = 0$

$z - 2 = 0$

Step 11.2

Set $z - 7$ equal to $0$ and solve for $z$ .

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Step 11.2.1

Set $z - 7$ equal to $0$ .

$z - 7 = 0$

Step 11.2.2

Add $7$ to both sides of the equation.

$z = 7$

Step 11.3

Set $z - 2$ equal to $0$ and solve for $z$ .

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Step 11.3.1

Set $z - 2$ equal to $0$ .

$z - 2 = 0$

Step 11.3.2

Add $2$ to both sides of the equation.

$z = 2$

Step 11.4

The final solution is all the values that make $(z - 7) (z - 2) = 0$ true.

$z = 7, 2$