Algebra Examples

$y^{2} - x^{2} = 56$ , $2 x - y = 1$

Step 1

Solve for $x$ in $2 x - y = 1$ .

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Step 1.1

Add $y$ to both sides of the equation.

$2 x = 1 + y$

$y^{2} - x^{2} = 56$

Step 1.2

Divide each term in $2 x = 1 + y$ by $2$ and simplify.

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Step 1.2.1

Divide each term in $2 x = 1 + y$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2} + \frac{y}{2}$

$y^{2} - x^{2} = 56$

Step 1.2.2

Simplify the left side.

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Step 1.2.2.1

Cancel the common factor of $2$ .

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Step 1.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2} + \frac{y}{2}$

$y^{2} - x^{2} = 56$

Step 1.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - x^{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - x^{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - x^{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - x^{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - x^{2} = 56$

Step 2

Replace all occurrences of $x$ with $\frac{1}{2} + \frac{y}{2}$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $y^{2} - x^{2} = 56$ with $\frac{1}{2} + \frac{y}{2}$ .

$y^{2} - {(\frac{1}{2} + \frac{y}{2})}^{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify $y^{2} - {(\frac{1}{2} + \frac{y}{2})}^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

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

$y^{2} - ((\frac{1}{2} + \frac{y}{2}) (\frac{1}{2} + \frac{y}{2})) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.2

Expand $(\frac{1}{2} + \frac{y}{2}) (\frac{1}{2} + \frac{y}{2})$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$y^{2} - (\frac{1}{2} \cdot (\frac{1}{2} + \frac{y}{2}) + \frac{y}{2} \cdot (\frac{1}{2} + \frac{y}{2})) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.2.2

Apply the distributive property.

$y^{2} - (\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot (\frac{1}{2} + \frac{y}{2})) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.2.3

Apply the distributive property.

$y^{2} - (\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{2} \cdot \frac{1}{2} + \frac{1}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3

Simplify and combine like terms.

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Step 2.2.1.1.3.1

Simplify each term.

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Step 2.2.1.1.3.1.1

Multiply $\frac{1}{2} \cdot \frac{1}{2}$ .

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Step 2.2.1.1.3.1.1.1

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

$y^{2} - (\frac{1}{2 \cdot 2} + \frac{1}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.1.2

Multiply $2$ by $2$ .

$y^{2} - (\frac{1}{4} + \frac{1}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{4} + \frac{1}{2} \cdot \frac{y}{2} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.2

Multiply $\frac{1}{2} \cdot \frac{y}{2}$ .

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Step 2.2.1.1.3.1.2.1

Multiply $\frac{1}{2}$ by $\frac{y}{2}$ .

$y^{2} - (\frac{1}{4} + \frac{y}{2 \cdot 2} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.2.2

Multiply $2$ by $2$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{2} \cdot \frac{1}{2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.3

Multiply $\frac{y}{2} \cdot \frac{1}{2}$ .

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Step 2.2.1.1.3.1.3.1

Multiply $\frac{y}{2}$ by $\frac{1}{2}$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{2 \cdot 2} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.3.2

Multiply $2$ by $2$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y}{2} \cdot \frac{y}{2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.4

Multiply $\frac{y}{2} \cdot \frac{y}{2}$ .

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Step 2.2.1.1.3.1.4.1

Multiply $\frac{y}{2}$ by $\frac{y}{2}$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y \cdot y}{2 \cdot 2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.4.2

Raise $y$ to the power of $1$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y \cdot y}{2 \cdot 2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.4.3

Raise $y$ to the power of $1$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y \cdot y}{2 \cdot 2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y^{1 + 1}}{2 \cdot 2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.4.5

Add $1$ and $1$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y^{2}}{2 \cdot 2}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.1.4.6

Multiply $2$ by $2$ .

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{4} + \frac{y}{4} + \frac{y}{4} + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.3.2

Add $\frac{y}{4}$ and $\frac{y}{4}$ .

$y^{2} - (\frac{1}{4} + 2 (\frac{y}{4}) + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{4} + 2 (\frac{y}{4}) + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.4

Cancel the common factor of $2$ .

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Step 2.2.1.1.4.1

Factor $2$ out of $4$ .

$y^{2} - (\frac{1}{4} + 2 (\frac{y}{2 (2)}) + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.4.2

Cancel the common factor.

$y^{2} - (\frac{1}{4} + 2 (\frac{y}{2 \cdot 2}) + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.4.3

Rewrite the expression.

$y^{2} - (\frac{1}{4} + \frac{y}{2} + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - (\frac{1}{4} + \frac{y}{2} + \frac{y^{2}}{4}) = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.1.5

Apply the distributive property.

$y^{2} - \frac{1}{4} - \frac{y}{2} - \frac{y^{2}}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$y^{2} - \frac{1}{4} - \frac{y}{2} - \frac{y^{2}}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.2

To write $y^{2}$ as a fraction with a common denominator, multiply by $\frac{4}{4}$ .

$y^{2} \cdot \frac{4}{4} - \frac{y^{2}}{4} - \frac{1}{4} - \frac{y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.3

Simplify terms.

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Step 2.2.1.3.1

Combine $y^{2}$ and $\frac{4}{4}$ .

$\frac{y^{2} \cdot 4}{4} - \frac{y^{2}}{4} - \frac{1}{4} - \frac{y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.3.2

Combine the numerators over the common denominator.

$\frac{y^{2} \cdot 4 - y^{2}}{4} - \frac{1}{4} - \frac{y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.3.3

Combine the numerators over the common denominator.

$\frac{y^{2} \cdot 4 - y^{2} - 1}{4} + \frac{- y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{y^{2} \cdot 4 - y^{2} - 1}{4} + \frac{- y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.4

Move $4$ to the left of $y^{2}$ .

$\frac{4 y^{2} - y^{2} - 1}{4} + \frac{- y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.5

Simplify by adding terms.

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Step 2.2.1.5.1

Subtract $y^{2}$ from $4 y^{2}$ .

$\frac{3 y^{2} - 1}{4} + \frac{- y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.5.2

Move the negative in front of the fraction.

$\frac{3 y^{2} - 1}{4} - \frac{y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{3 y^{2} - 1}{4} - \frac{y}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.6

To write $- \frac{y}{2}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{3 y^{2} - 1}{4} - \frac{y}{2} \cdot \frac{2}{2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.7

Write each expression with a common denominator of $4$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.2.1.7.1

Multiply $\frac{y}{2}$ by $\frac{2}{2}$ .

$\frac{3 y^{2} - 1}{4} - \frac{y \cdot 2}{2 \cdot 2} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.7.2

Multiply $2$ by $2$ .

$\frac{3 y^{2} - 1}{4} - \frac{y \cdot 2}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{3 y^{2} - 1}{4} - \frac{y \cdot 2}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.8

Combine the numerators over the common denominator.

$\frac{3 y^{2} - 1 - y \cdot 2}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9

Simplify the numerator.

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Step 2.2.1.9.1

Multiply $2$ by $- 1$ .

$\frac{3 y^{2} - 1 - 2 y}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.2

Reorder terms.

$\frac{3 y^{2} - 2 y - 1}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.3

Factor by grouping.

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Step 2.2.1.9.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 1 = - 3$ and whose sum is $b = - 2$ .

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Step 2.2.1.9.3.1.1

Factor $- 2$ out of $- 2 y$ .

$\frac{3 y^{2} - 2 y - 1}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.3.1.2

Rewrite $- 2$ as $1$ plus $- 3$

$\frac{3 y^{2} + (1 - 3) y - 1}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.3.1.3

Apply the distributive property.

$\frac{3 y^{2} + 1 y - 3 y - 1}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.3.1.4

Multiply $y$ by $1$ .

$\frac{3 y^{2} + y - 3 y - 1}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{3 y^{2} + y - 3 y - 1}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.3.2

Factor out the greatest common factor from each group.

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Step 2.2.1.9.3.2.1

Group the first two terms and the last two terms.

$\frac{(3 y^{2} + y) - 3 y - 1}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.3.2.2

Factor out the greatest common factor (GCF) from each group.

$\frac{y (3 y + 1) - (3 y + 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{y (3 y + 1) - (3 y + 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 2.2.1.9.3.3

Factor the polynomial by factoring out the greatest common factor, $3 y + 1$ .

$\frac{(3 y + 1) (y - 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{(3 y + 1) (y - 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{(3 y + 1) (y - 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{(3 y + 1) (y - 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{(3 y + 1) (y - 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

$\frac{(3 y + 1) (y - 1)}{4} = 56$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3

Solve for $y$ in $\frac{(3 y + 1) (y - 1)}{4} = 56$ .

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Step 3.1

Multiply both sides by $4$ .

$\frac{(3 y + 1) (y - 1)}{4} \cdot 4 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2

Simplify.

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Step 3.2.1

Simplify the left side.

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Step 3.2.1.1

Simplify $\frac{(3 y + 1) (y - 1)}{4} \cdot 4$ .

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Step 3.2.1.1.1

Cancel the common factor of $4$ .

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Step 3.2.1.1.1.1

Cancel the common factor.

$\frac{(3 y + 1) (y - 1)}{4} \cdot 4 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.1.2

Rewrite the expression.

$(3 y + 1) (y - 1) = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

$(3 y + 1) (y - 1) = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.2

Expand $(3 y + 1) (y - 1)$ using the FOIL Method.

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Step 3.2.1.1.2.1

Apply the distributive property.

$3 y (y - 1) + 1 (y - 1) = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.2.2

Apply the distributive property.

$3 y \cdot y + 3 y \cdot - 1 + 1 (y - 1) = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.2.3

Apply the distributive property.

$3 y \cdot y + 3 y \cdot - 1 + 1 y + 1 \cdot - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y \cdot y + 3 y \cdot - 1 + 1 y + 1 \cdot - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.3

Simplify and combine like terms.

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Step 3.2.1.1.3.1

Simplify each term.

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Step 3.2.1.1.3.1.1

Multiply $y$ by $y$ by adding the exponents.

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Step 3.2.1.1.3.1.1.1

Move $y$ .

$3 (y \cdot y) + 3 y \cdot - 1 + 1 y + 1 \cdot - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.3.1.1.2

Multiply $y$ by $y$ .

$3 y^{2} + 3 y \cdot - 1 + 1 y + 1 \cdot - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} + 3 y \cdot - 1 + 1 y + 1 \cdot - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.3.1.2

Multiply $- 1$ by $3$ .

$3 y^{2} - 3 y + 1 y + 1 \cdot - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.3.1.3

Multiply $y$ by $1$ .

$3 y^{2} - 3 y + y + 1 \cdot - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.3.1.4

Multiply $- 1$ by $1$ .

$3 y^{2} - 3 y + y - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} - 3 y + y - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.1.1.3.2

Add $- 3 y$ and $y$ .

$3 y^{2} - 2 y - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} - 2 y - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} - 2 y - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} - 2 y - 1 = 56 \cdot 4$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.2.2

Simplify the right side.

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Step 3.2.2.1

Multiply $56$ by $4$ .

$3 y^{2} - 2 y - 1 = 224$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} - 2 y - 1 = 224$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} - 2 y - 1 = 224$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3

Solve for $y$ .

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Step 3.3.1

Subtract $224$ from both sides of the equation.

$3 y^{2} - 2 y - 1 - 224 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.2

Subtract $224$ from $- 1$ .

$3 y^{2} - 2 y - 225 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.3

Factor by grouping.

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Step 3.3.3.1

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 3 \cdot - 225 = - 675$ and whose sum is $b = - 2$ .

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Step 3.3.3.1.1

Factor $- 2$ out of $- 2 y$ .

$3 y^{2} - 2 y - 225 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.3.1.2

Rewrite $- 2$ as $25$ plus $- 27$

$3 y^{2} + (25 - 27) y - 225 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.3.1.3

Apply the distributive property.

$3 y^{2} + 25 y - 27 y - 225 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

$3 y^{2} + 25 y - 27 y - 225 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.3.2

Factor out the greatest common factor from each group.

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Step 3.3.3.2.1

Group the first two terms and the last two terms.

$(3 y^{2} + 25 y) - 27 y - 225 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.3.2.2

Factor out the greatest common factor (GCF) from each group.

$y (3 y + 25) - 9 (3 y + 25) = 0$

$x = \frac{1}{2} + \frac{y}{2}$

$y (3 y + 25) - 9 (3 y + 25) = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.3.3

Factor the polynomial by factoring out the greatest common factor, $3 y + 25$ .

$(3 y + 25) (y - 9) = 0$

$x = \frac{1}{2} + \frac{y}{2}$

$(3 y + 25) (y - 9) = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.4

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

$3 y + 25 = 0$

$y - 9 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.5

Set $3 y + 25$ equal to $0$ and solve for $y$ .

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Step 3.3.5.1

Set $3 y + 25$ equal to $0$ .

$3 y + 25 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.5.2

Solve $3 y + 25 = 0$ for $y$ .

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Step 3.3.5.2.1

Subtract $25$ from both sides of the equation.

$3 y = - 25$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.5.2.2

Divide each term in $3 y = - 25$ by $3$ and simplify.

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Step 3.3.5.2.2.1

Divide each term in $3 y = - 25$ by $3$ .

$\frac{3 y}{3} = \frac{- 25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.5.2.2.2

Simplify the left side.

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Step 3.3.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 3.3.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 y}{3} = \frac{- 25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.5.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

$y = \frac{- 25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

$y = \frac{- 25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.5.2.2.3

Simplify the right side.

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Step 3.3.5.2.2.3.1

Move the negative in front of the fraction.

$y = - \frac{25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

$y = - \frac{25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

$y = - \frac{25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

$y = - \frac{25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

$y = - \frac{25}{3}$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.6

Set $y - 9$ equal to $0$ and solve for $y$ .

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Step 3.3.6.1

Set $y - 9$ equal to $0$ .

$y - 9 = 0$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.6.2

Add $9$ to both sides of the equation.

$y = 9$

$x = \frac{1}{2} + \frac{y}{2}$

$y = 9$

$x = \frac{1}{2} + \frac{y}{2}$

Step 3.3.7

The final solution is all the values that make $(3 y + 25) (y - 9) = 0$ true.

$y = - \frac{25}{3}, 9$

$x = \frac{1}{2} + \frac{y}{2}$

$y = - \frac{25}{3}, 9$

$x = \frac{1}{2} + \frac{y}{2}$

$y = - \frac{25}{3}, 9$

$x = \frac{1}{2} + \frac{y}{2}$

Step 4

Replace all occurrences of $y$ with $- \frac{25}{3}$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = \frac{1}{2} + \frac{y}{2}$ with $- \frac{25}{3}$ .

$x = \frac{1}{2} + \frac{- \frac{25}{3}}{2}$

$y = - \frac{25}{3}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{1}{2} + \frac{- \frac{25}{3}}{2}$ .

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Step 4.2.1.1

Combine the numerators over the common denominator.

$x = \frac{1 - \frac{25}{3}}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.2

Simplify the expression.

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Step 4.2.1.2.1

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

$x = \frac{\frac{3}{3} - \frac{25}{3}}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.2.2

Combine the numerators over the common denominator.

$x = \frac{\frac{3 - 25}{3}}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.2.3

Subtract $25$ from $3$ .

$x = \frac{\frac{- 22}{3}}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.2.4

Move the negative in front of the fraction.

$x = \frac{- \frac{22}{3}}{2}$

$y = - \frac{25}{3}$

$x = \frac{- \frac{22}{3}}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.3

Multiply the numerator by the reciprocal of the denominator.

$x = - \frac{22}{3} \cdot \frac{1}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.4

Cancel the common factor of $2$ .

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Step 4.2.1.4.1

Move the leading negative in $- \frac{22}{3}$ into the numerator.

$x = \frac{- 22}{3} \cdot \frac{1}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.4.2

Factor $2$ out of $- 22$ .

$x = \frac{2 (- 11)}{3} \cdot \frac{1}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.4.3

Cancel the common factor.

$x = \frac{2 \cdot - 11}{3} \cdot \frac{1}{2}$

$y = - \frac{25}{3}$

Step 4.2.1.4.4

Rewrite the expression.

$x = \frac{- 11}{3}$

$y = - \frac{25}{3}$

$x = \frac{- 11}{3}$

$y = - \frac{25}{3}$

Step 4.2.1.5

Move the negative in front of the fraction.

$x = - \frac{11}{3}$

$y = - \frac{25}{3}$

$x = - \frac{11}{3}$

$y = - \frac{25}{3}$

$x = - \frac{11}{3}$

$y = - \frac{25}{3}$

$x = - \frac{11}{3}$

$y = - \frac{25}{3}$

Step 5

Replace all occurrences of $y$ with $9$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = \frac{1}{2} + \frac{y}{2}$ with $9$ .

$x = \frac{1}{2} + \frac{9}{2}$

$y = 9$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $\frac{1}{2} + \frac{9}{2}$ .

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Step 5.2.1.1

Combine the numerators over the common denominator.

$x = \frac{1 + 9}{2}$

$y = 9$

Step 5.2.1.2

Simplify the expression.

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Step 5.2.1.2.1

Add $1$ and $9$ .

$x = \frac{10}{2}$

$y = 9$

Step 5.2.1.2.2

Divide $10$ by $2$ .

$x = 5$

$y = 9$

$x = 5$

$y = 9$

$x = 5$

$y = 9$

$x = 5$

$y = 9$

$x = 5$

$y = 9$

Step 6

The solution to the system is the complete set of ordered pairs that are valid solutions.

$(- \frac{11}{3}, - \frac{25}{3})$

$(5, 9)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- \frac{11}{3}, - \frac{25}{3}), (5, 9)$

Equation Form:

$x = - \frac{11}{3}, y = - \frac{25}{3}$

$x = 5, y = 9$

Step 8