Basic Math Examples

\frac{1}{3} \cdot (y - 9) = - 4 y + \frac{13}{3} y - 3

$\frac{1}{3} \cdot (y - 9) = - 4 y + \frac{13}{3} y - 3$

Step 1

Since

y

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

- 4 y + \frac{13}{3} y - 3 = \frac{1}{3} \cdot (y - 9)

$- 4 y + \frac{13}{3} y - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2

Simplify

- 4 y + \frac{13}{3} y - 3

$- 4 y + \frac{13}{3} y - 3$ .

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Step 2.1

Combine

\frac{13}{3}

$\frac{13}{3}$ and

y

$y$ .

- 4 y + \frac{13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$- 4 y + \frac{13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.2

To write

- 4 y

$- 4 y$ as a fraction with a common denominator, multiply by

\frac{3}{3}

$\frac{3}{3}$ .

- 4 y \cdot \frac{3}{3} + \frac{13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$- 4 y \cdot \frac{3}{3} + \frac{13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.3

Simplify terms.

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Step 2.3.1

Combine

- 4 y

$- 4 y$ and

\frac{3}{3}

$\frac{3}{3}$ .

\frac{- 4 y \cdot 3}{3} + \frac{13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{- 4 y \cdot 3}{3} + \frac{13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.3.2

Combine the numerators over the common denominator.

\frac{- 4 y \cdot 3 + 13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{- 4 y \cdot 3 + 13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

\frac{- 4 y \cdot 3 + 13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{- 4 y \cdot 3 + 13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.4

Simplify each term.

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Step 2.4.1

Simplify the numerator.

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Step 2.4.1.1

Factor

y

$y$ out of

- 4 y \cdot 3 + 13 y

$- 4 y \cdot 3 + 13 y$ .

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Step 2.4.1.1.1

Factor

y

$y$ out of

- 4 y \cdot 3

$- 4 y \cdot 3$ .

\frac{y (- 4 \cdot 3) + 13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y (- 4 \cdot 3) + 13 y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.4.1.1.2

Factor

y

$y$ out of

13 y

$13 y$ .

\frac{y (- 4 \cdot 3) + y \cdot 13}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y (- 4 \cdot 3) + y \cdot 13}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.4.1.1.3

Factor

y

$y$ out of

y (- 4 \cdot 3) + y \cdot 13

$y (- 4 \cdot 3) + y \cdot 13$ .

\frac{y (- 4 \cdot 3 + 13)}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y (- 4 \cdot 3 + 13)}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

\frac{y (- 4 \cdot 3 + 13)}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y (- 4 \cdot 3 + 13)}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.4.1.2

Multiply

- 4

$- 4$ by

3

$3$ .

\frac{y (- 12 + 13)}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y (- 12 + 13)}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.4.1.3

Add

- 12

$- 12$ and

13

$13$ .

\frac{y \cdot 1}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y \cdot 1}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

\frac{y \cdot 1}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y \cdot 1}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 2.4.2

Multiply

y

$y$ by

1

$1$ .

\frac{y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

\frac{y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

\frac{y}{3} - 3 = \frac{1}{3} \cdot (y - 9)

$\frac{y}{3} - 3 = \frac{1}{3} \cdot (y - 9)$

Step 3

Simplify

\frac{1}{3} \cdot (y - 9)

$\frac{1}{3} \cdot (y - 9)$ .

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Step 3.1

Apply the distributive property.

\frac{y}{3} - 3 = \frac{1}{3} y + \frac{1}{3} \cdot - 9

$\frac{y}{3} - 3 = \frac{1}{3} y + \frac{1}{3} \cdot - 9$

Step 3.2

Combine

\frac{1}{3}

$\frac{1}{3}$ and

y

$y$ .

\frac{y}{3} - 3 = \frac{y}{3} + \frac{1}{3} \cdot - 9

$\frac{y}{3} - 3 = \frac{y}{3} + \frac{1}{3} \cdot - 9$

Step 3.3

Cancel the common factor of

3

$3$ .

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Step 3.3.1

Factor

3

$3$ out of

- 9

$- 9$ .

\frac{y}{3} - 3 = \frac{y}{3} + \frac{1}{3} \cdot (3 (- 3))

$\frac{y}{3} - 3 = \frac{y}{3} + \frac{1}{3} \cdot (3 (- 3))$

Step 3.3.2

Cancel the common factor.

$\frac{y}{3} - 3 = \frac{y}{3} + \frac{1}{3} \cdot (3 \cdot - 3)$

Step 3.3.3

Rewrite the expression.

$\frac{y}{3} - 3 = \frac{y}{3} - 3$

Step 4

Move all terms containing

$y$ to the left side of the equation.

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Step 4.1

Subtract

$\frac{y}{3}$ from both sides of the equation.

$\frac{y}{3} - 3 - \frac{y}{3} = - 3$

Step 4.2

Combine the opposite terms in

$\frac{y}{3} - 3 - \frac{y}{3}$ .

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Step 4.2.1

Subtract

$\frac{y}{3}$ from

$\frac{y}{3}$ .

$0 - 3 = - 3$

Step 4.2.2

Subtract

$3$ from

$0$ .

$- 3 = - 3$

Step 5

Since

$- 3 = - 3$ , the equation will always be true for any value of

$y$ .

All real numbers

Step 6

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$