Basic Math Examples

${(y - 7)}^{2} - {(1 + y)}^{2} = 2 \cdot (3 y - 4)$

Step 1

Simplify $2 \cdot (3 y - 4)$ .

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Step 1.1

Apply the distributive property.

${(y - 7)}^{2} - {(1 + y)}^{2} = 2 (3 y) + 2 \cdot - 4$

Step 1.2

Multiply.

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Step 1.2.1

Multiply $3$ by $2$ .

${(y - 7)}^{2} - {(1 + y)}^{2} = 6 y + 2 \cdot - 4$

Step 1.2.2

Multiply $2$ by $- 4$ .

${(y - 7)}^{2} - {(1 + y)}^{2} = 6 y - 8$

Step 2

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

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Step 2.1

Subtract $6 y$ from both sides of the equation.

${(y - 7)}^{2} - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2

Simplify each term.

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Step 2.2.1

Rewrite ${(y - 7)}^{2}$ as $(y - 7) (y - 7)$ .

$(y - 7) (y - 7) - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.2

Expand $(y - 7) (y - 7)$ using the FOIL Method.

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Step 2.2.2.1

Apply the distributive property.

$y (y - 7) - 7 (y - 7) - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.2.2

Apply the distributive property.

$y \cdot y + y \cdot - 7 - 7 (y - 7) - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.2.3

Apply the distributive property.

$y \cdot y + y \cdot - 7 - 7 y - 7 \cdot - 7 - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.3

Simplify and combine like terms.

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Step 2.2.3.1

Simplify each term.

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Step 2.2.3.1.1

Multiply $y$ by $y$ .

$y^{2} + y \cdot - 7 - 7 y - 7 \cdot - 7 - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.3.1.2

Move $- 7$ to the left of $y$ .

$y^{2} - 7 \cdot y - 7 y - 7 \cdot - 7 - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.3.1.3

Multiply $- 7$ by $- 7$ .

$y^{2} - 7 y - 7 y + 49 - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.3.2

Subtract $7 y$ from $- 7 y$ .

$y^{2} - 14 y + 49 - {(1 + y)}^{2} - 6 y = - 8$

Step 2.2.4

Rewrite ${(1 + y)}^{2}$ as $(1 + y) (1 + y)$ .

$y^{2} - 14 y + 49 - ((1 + y) (1 + y)) - 6 y = - 8$

Step 2.2.5

Expand $(1 + y) (1 + y)$ using the FOIL Method.

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Step 2.2.5.1

Apply the distributive property.

$y^{2} - 14 y + 49 - (1 (1 + y) + y (1 + y)) - 6 y = - 8$

Step 2.2.5.2

Apply the distributive property.

$y^{2} - 14 y + 49 - (1 \cdot 1 + 1 y + y (1 + y)) - 6 y = - 8$

Step 2.2.5.3

Apply the distributive property.

$y^{2} - 14 y + 49 - (1 \cdot 1 + 1 y + y \cdot 1 + y \cdot y) - 6 y = - 8$

Step 2.2.6

Simplify and combine like terms.

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Step 2.2.6.1

Simplify each term.

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Step 2.2.6.1.1

Multiply $1$ by $1$ .

$y^{2} - 14 y + 49 - (1 + 1 y + y \cdot 1 + y \cdot y) - 6 y = - 8$

Step 2.2.6.1.2

Multiply $y$ by $1$ .

$y^{2} - 14 y + 49 - (1 + y + y \cdot 1 + y \cdot y) - 6 y = - 8$

Step 2.2.6.1.3

Multiply $y$ by $1$ .

$y^{2} - 14 y + 49 - (1 + y + y + y \cdot y) - 6 y = - 8$

Step 2.2.6.1.4

Multiply $y$ by $y$ .

$y^{2} - 14 y + 49 - (1 + y + y + y^{2}) - 6 y = - 8$

Step 2.2.6.2

Add $y$ and $y$ .

$y^{2} - 14 y + 49 - (1 + 2 y + y^{2}) - 6 y = - 8$

Step 2.2.7

Apply the distributive property.

$y^{2} - 14 y + 49 - 1 \cdot 1 - (2 y) - y^{2} - 6 y = - 8$

Step 2.2.8

Simplify.

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Step 2.2.8.1

Multiply $- 1$ by $1$ .

$y^{2} - 14 y + 49 - 1 - (2 y) - y^{2} - 6 y = - 8$

Step 2.2.8.2

Multiply $2$ by $- 1$ .

$y^{2} - 14 y + 49 - 1 - 2 y - y^{2} - 6 y = - 8$

Step 2.3

Combine the opposite terms in $y^{2} - 14 y + 49 - 1 - 2 y - y^{2} - 6 y$ .

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Step 2.3.1

Subtract $y^{2}$ from $y^{2}$ .

$- 14 y + 49 - 1 - 2 y + 0 - 6 y = - 8$

Step 2.3.2

Add $- 14 y + 49 - 1 - 2 y$ and $0$ .

$- 14 y + 49 - 1 - 2 y - 6 y = - 8$

Step 2.4

Subtract $2 y$ from $- 14 y$ .

$- 16 y + 49 - 1 - 6 y = - 8$

Step 2.5

Subtract $6 y$ from $- 16 y$ .

$- 22 y + 49 - 1 = - 8$

Step 2.6

Subtract $1$ from $49$ .

$- 22 y + 48 = - 8$

Step 3

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

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Step 3.1

Subtract $48$ from both sides of the equation.

$- 22 y = - 8 - 48$

Step 3.2

Subtract $48$ from $- 8$ .

$- 22 y = - 56$

Step 4

Divide each term in $- 22 y = - 56$ by $- 22$ and simplify.

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Step 4.1

Divide each term in $- 22 y = - 56$ by $- 22$ .

$\frac{- 22 y}{- 22} = \frac{- 56}{- 22}$

Step 4.2

Simplify the left side.

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Step 4.2.1

Cancel the common factor of $- 22$ .

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Step 4.2.1.1

Cancel the common factor.

$\frac{- 22 y}{- 22} = \frac{- 56}{- 22}$

Step 4.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 56}{- 22}$

Step 4.3

Simplify the right side.

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Step 4.3.1

Cancel the common factor of $- 56$ and $- 22$ .

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Step 4.3.1.1

Factor $- 2$ out of $- 56$ .

$y = \frac{- 2 (28)}{- 22}$

Step 4.3.1.2

Cancel the common factors.

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Step 4.3.1.2.1

Factor $- 2$ out of $- 22$ .

$y = \frac{- 2 \cdot 28}{- 2 \cdot 11}$

Step 4.3.1.2.2

Cancel the common factor.

$y = \frac{- 2 \cdot 28}{- 2 \cdot 11}$

Step 4.3.1.2.3

Rewrite the expression.

$y = \frac{28}{11}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{28}{11}$

Decimal Form:

$y = 2.\overline{54}$

Mixed Number Form:

$y = 2 \frac{6}{11}$