Algebra Examples

$x^{2} - 4 y^{2} = 16$ $x - 6 y = - 4$

Step 1

Add $6 y$ to both sides of the equation.

$x = - 4 + 6 y$

$x^{2} - 4 y^{2} = 16$

Step 2

Replace all occurrences of $x$ with $- 4 + 6 y$ in each equation.

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Step 2.1

Replace all occurrences of $x$ in $x^{2} - 4 y^{2} = 16$ with $- 4 + 6 y$ .

${(- 4 + 6 y)}^{2} - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2

Simplify the left side.

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Step 2.2.1

Simplify ${(- 4 + 6 y)}^{2} - 4 y^{2}$ .

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Step 2.2.1.1

Simplify each term.

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Step 2.2.1.1.1

Rewrite ${(- 4 + 6 y)}^{2}$ as $(- 4 + 6 y) (- 4 + 6 y)$ .

$(- 4 + 6 y) (- 4 + 6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.2

Expand $(- 4 + 6 y) (- 4 + 6 y)$ using the FOIL Method.

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Step 2.2.1.1.2.1

Apply the distributive property.

$- 4 (- 4 + 6 y) + 6 y (- 4 + 6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.2.2

Apply the distributive property.

$- 4 \cdot - 4 - 4 (6 y) + 6 y (- 4 + 6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.2.3

Apply the distributive property.

$- 4 \cdot - 4 - 4 (6 y) + 6 y \cdot - 4 + 6 y (6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

$- 4 \cdot - 4 - 4 (6 y) + 6 y \cdot - 4 + 6 y (6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

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 $- 4$ by $- 4$ .

$16 - 4 (6 y) + 6 y \cdot - 4 + 6 y (6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.3.1.2

Multiply $6$ by $- 4$ .

$16 - 24 y + 6 y \cdot - 4 + 6 y (6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.3.1.3

Multiply $- 4$ by $6$ .

$16 - 24 y - 24 y + 6 y (6 y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$16 - 24 y - 24 y + 6 \cdot (6 y \cdot y) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.3.1.5

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

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Step 2.2.1.1.3.1.5.1

Move $y$ .

$16 - 24 y - 24 y + 6 \cdot (6 (y \cdot y)) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$16 - 24 y - 24 y + 6 \cdot (6 y^{2}) - 4 y^{2} = 16$

$x = - 4 + 6 y$

$16 - 24 y - 24 y + 6 \cdot (6 y^{2}) - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.3.1.6

Multiply $6$ by $6$ .

$16 - 24 y - 24 y + 36 y^{2} - 4 y^{2} = 16$

$x = - 4 + 6 y$

$16 - 24 y - 24 y + 36 y^{2} - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.1.3.2

Subtract $24 y$ from $- 24 y$ .

$16 - 48 y + 36 y^{2} - 4 y^{2} = 16$

$x = - 4 + 6 y$

$16 - 48 y + 36 y^{2} - 4 y^{2} = 16$

$x = - 4 + 6 y$

$16 - 48 y + 36 y^{2} - 4 y^{2} = 16$

$x = - 4 + 6 y$

Step 2.2.1.2

Subtract $4 y^{2}$ from $36 y^{2}$ .

$16 - 48 y + 32 y^{2} = 16$

$x = - 4 + 6 y$

$16 - 48 y + 32 y^{2} = 16$

$x = - 4 + 6 y$

$16 - 48 y + 32 y^{2} = 16$

$x = - 4 + 6 y$

$16 - 48 y + 32 y^{2} = 16$

$x = - 4 + 6 y$

Step 3

Solve for $y$ in $16 - 48 y + 32 y^{2} = 16$ .

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Step 3.1

Subtract $16$ from both sides of the equation.

$16 - 48 y + 32 y^{2} - 16 = 0$

$x = - 4 + 6 y$

Step 3.2

Combine the opposite terms in $16 - 48 y + 32 y^{2} - 16$ .

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Step 3.2.1

Subtract $16$ from $16$ .

$- 48 y + 32 y^{2} + 0 = 0$

$x = - 4 + 6 y$

Step 3.2.2

Add $- 48 y + 32 y^{2}$ and $0$ .

$- 48 y + 32 y^{2} = 0$

$x = - 4 + 6 y$

$- 48 y + 32 y^{2} = 0$

$x = - 4 + 6 y$

Step 3.3

Factor $- 16 y$ out of $- 48 y + 32 y^{2}$ .

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Step 3.3.1

Reorder $- 48 y$ and $32 y^{2}$ .

$32 y^{2} - 48 y = 0$

$x = - 4 + 6 y$

Step 3.3.2

Factor $- 16 y$ out of $32 y^{2}$ .

$- 16 y (- 2 y) - 48 y = 0$

$x = - 4 + 6 y$

Step 3.3.3

Factor $- 16 y$ out of $- 48 y$ .

$- 16 y (- 2 y) - 16 y \cdot 3 = 0$

$x = - 4 + 6 y$

Step 3.3.4

Factor $- 16 y$ out of $- 16 y (- 2 y) - 16 y (3)$ .

$- 16 y (- 2 y + 3) = 0$

$x = - 4 + 6 y$

$- 16 y (- 2 y + 3) = 0$

$x = - 4 + 6 y$

Step 3.4

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

$y = 0$

$- 2 y + 3 = 0$

$x = - 4 + 6 y$

Step 3.5

Set $y$ equal to $0$ .

$y = 0$

$x = - 4 + 6 y$

Step 3.6

Set $- 2 y + 3$ equal to $0$ and solve for $y$ .

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Step 3.6.1

Set $- 2 y + 3$ equal to $0$ .

$- 2 y + 3 = 0$

$x = - 4 + 6 y$

Step 3.6.2

Solve $- 2 y + 3 = 0$ for $y$ .

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Step 3.6.2.1

Subtract $3$ from both sides of the equation.

$- 2 y = - 3$

$x = - 4 + 6 y$

Step 3.6.2.2

Divide each term in $- 2 y = - 3$ by $- 2$ and simplify.

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Step 3.6.2.2.1

Divide each term in $- 2 y = - 3$ by $- 2$ .

$\frac{- 2 y}{- 2} = \frac{- 3}{- 2}$

$x = - 4 + 6 y$

Step 3.6.2.2.2

Simplify the left side.

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Step 3.6.2.2.2.1

Cancel the common factor of $- 2$ .

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Step 3.6.2.2.2.1.1

Cancel the common factor.

$\frac{- 2 y}{- 2} = \frac{- 3}{- 2}$

$x = - 4 + 6 y$

Step 3.6.2.2.2.1.2

Divide $y$ by $1$ .

$y = \frac{- 3}{- 2}$

$x = - 4 + 6 y$

$y = \frac{- 3}{- 2}$

$x = - 4 + 6 y$

$y = \frac{- 3}{- 2}$

$x = - 4 + 6 y$

Step 3.6.2.2.3

Simplify the right side.

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Step 3.6.2.2.3.1

Dividing two negative values results in a positive value.

$y = \frac{3}{2}$

$x = - 4 + 6 y$

$y = \frac{3}{2}$

$x = - 4 + 6 y$

$y = \frac{3}{2}$

$x = - 4 + 6 y$

$y = \frac{3}{2}$

$x = - 4 + 6 y$

$y = \frac{3}{2}$

$x = - 4 + 6 y$

Step 3.7

The final solution is all the values that make $- 16 y (- 2 y + 3) = 0$ true.

$y = 0, \frac{3}{2}$

$x = - 4 + 6 y$

$y = 0, \frac{3}{2}$

$x = - 4 + 6 y$

Step 4

Replace all occurrences of $y$ with $0$ in each equation.

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Step 4.1

Replace all occurrences of $y$ in $x = - 4 + 6 y$ with $0$ .

$x = - 4 + 6 (0)$

$y = 0$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $- 4 + 6 (0)$ .

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Step 4.2.1.1

Multiply $6$ by $0$ .

$x = - 4 + 0$

$y = 0$

Step 4.2.1.2

Add $- 4$ and $0$ .

$x = - 4$

$y = 0$

$x = - 4$

$y = 0$

$x = - 4$

$y = 0$

$x = - 4$

$y = 0$

Step 5

Replace all occurrences of $y$ with $\frac{3}{2}$ in each equation.

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Step 5.1

Replace all occurrences of $y$ in $x = - 4 + 6 y$ with $\frac{3}{2}$ .

$x = - 4 + 6 (\frac{3}{2})$

$y = \frac{3}{2}$

Step 5.2

Simplify the right side.

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Step 5.2.1

Simplify $- 4 + 6 (\frac{3}{2})$ .

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Step 5.2.1.1

Simplify each term.

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Step 5.2.1.1.1

Cancel the common factor of $2$ .

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Step 5.2.1.1.1.1

Factor $2$ out of $6$ .

$x = - 4 + 2 (3) (\frac{3}{2})$

$y = \frac{3}{2}$

Step 5.2.1.1.1.2

Cancel the common factor.

$x = - 4 + 2 \cdot (3 (\frac{3}{2}))$

$y = \frac{3}{2}$

Step 5.2.1.1.1.3

Rewrite the expression.

$x = - 4 + 3 \cdot 3$

$y = \frac{3}{2}$

$x = - 4 + 3 \cdot 3$

$y = \frac{3}{2}$

Step 5.2.1.1.2

Multiply $3$ by $3$ .

$x = - 4 + 9$

$y = \frac{3}{2}$

$x = - 4 + 9$

$y = \frac{3}{2}$

Step 5.2.1.2

Add $- 4$ and $9$ .

$x = 5$

$y = \frac{3}{2}$

$x = 5$

$y = \frac{3}{2}$

$x = 5$

$y = \frac{3}{2}$

$x = 5$

$y = \frac{3}{2}$

Step 6

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

$(- 4, 0)$

$(5, \frac{3}{2})$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 4, 0), (5, \frac{3}{2})$

Equation Form:

$x = - 4, y = 0$

$x = 5, y = \frac{3}{2}$

Step 8