Solve the System of Equations x^2-4y^2=36 x-y=6

$x^{2} - 4 y^{2} = 36$ $x - y = 6$

Step 1

Add $y$ to both sides of the equation.

$x = 6 + y$

$x^{2} - 4 y^{2} = 36$

Step 2

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

Replace all occurrences of $x$ in $x^{2} - 4 y^{2} = 36$ with $6 + y$ .

${(6 + y)}^{2} - 4 y^{2} = 36$

$x = 6 + y$

Simplify the left side.

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

Simplify each term.

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

$(6 + y) (6 + y) - 4 y^{2} = 36$

$x = 6 + y$

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

Apply the distributive property.

$6 (6 + y) + y (6 + y) - 4 y^{2} = 36$

$x = 6 + y$

Apply the distributive property.

$6 \cdot 6 + 6 y + y (6 + y) - 4 y^{2} = 36$

$x = 6 + y$

Apply the distributive property.

$6 \cdot 6 + 6 y + y \cdot 6 + y \cdot y - 4 y^{2} = 36$

$x = 6 + y$

$6 \cdot 6 + 6 y + y \cdot 6 + y \cdot y - 4 y^{2} = 36$

$x = 6 + y$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$36 + 6 y + y \cdot 6 + y \cdot y - 4 y^{2} = 36$

$x = 6 + y$

Move $6$ to the left of $y$ .

$36 + 6 y + 6 \cdot y + y \cdot y - 4 y^{2} = 36$

$x = 6 + y$

Multiply $y$ by $y$ .

$36 + 6 y + 6 y + y^{2} - 4 y^{2} = 36$

$x = 6 + y$

$36 + 6 y + 6 y + y^{2} - 4 y^{2} = 36$

$x = 6 + y$

Add $6 y$ and $6 y$ .

$36 + 12 y + y^{2} - 4 y^{2} = 36$

$x = 6 + y$

$36 + 12 y + y^{2} - 4 y^{2} = 36$

$x = 6 + y$

$36 + 12 y + y^{2} - 4 y^{2} = 36$

$x = 6 + y$

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

$36 + 12 y - 3 y^{2} = 36$

$x = 6 + y$

$36 + 12 y - 3 y^{2} = 36$

$x = 6 + y$

$36 + 12 y - 3 y^{2} = 36$

$x = 6 + y$

$36 + 12 y - 3 y^{2} = 36$

$x = 6 + y$

Step 3

Solve for $y$ in $36 + 12 y - 3 y^{2} = 36$ .

Subtract $36$ from both sides of the equation.

$36 + 12 y - 3 y^{2} - 36 = 0$

$x = 6 + y$

Combine the opposite terms in $36 + 12 y - 3 y^{2} - 36$ .

Subtract $36$ from $36$ .

$12 y - 3 y^{2} + 0 = 0$

$x = 6 + y$

Add $12 y - 3 y^{2}$ and $0$ .

$12 y - 3 y^{2} = 0$

$x = 6 + y$

$12 y - 3 y^{2} = 0$

$x = 6 + y$

Factor $3 y$ out of $12 y - 3 y^{2}$ .

Factor $3 y$ out of $12 y$ .

$3 y (4) - 3 y^{2} = 0$

$x = 6 + y$

Factor $3 y$ out of $- 3 y^{2}$ .

$3 y (4) + 3 y (- y) = 0$

$x = 6 + y$

Factor $3 y$ out of $3 y (4) + 3 y (- y)$ .

$3 y (4 - y) = 0$

$x = 6 + y$

$3 y (4 - y) = 0$

$x = 6 + y$

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

$y = 0$

$4 - y = 0$

$x = 6 + y$

Set $y$ equal to $0$ .

$y = 0$

$x = 6 + y$

Set $4 - y$ equal to $0$ and solve for $y$ .

Set $4 - y$ equal to $0$ .

$4 - y = 0$

$x = 6 + y$

Solve $4 - y = 0$ for $y$ .

Subtract $4$ from both sides of the equation.

$- y = - 4$

$x = 6 + y$

Divide each term in $- y = - 4$ by $- 1$ and simplify.

Divide each term in $- y = - 4$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 4}{- 1}$

$x = 6 + y$

Simplify the left side.

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 4}{- 1}$

$x = 6 + y$

Divide $y$ by $1$ .

$y = \frac{- 4}{- 1}$

$x = 6 + y$

$y = \frac{- 4}{- 1}$

$x = 6 + y$

Simplify the right side.

Divide $- 4$ by $- 1$ .

$y = 4$

$x = 6 + y$

$y = 4$

$x = 6 + y$

$y = 4$

$x = 6 + y$

$y = 4$

$x = 6 + y$

$y = 4$

$x = 6 + y$

The final solution is all the values that make $3 y (4 - y) = 0$ true.

$y = 0, 4$

$x = 6 + y$

$y = 0, 4$

$x = 6 + y$

Step 4

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

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

$x = 6 + 0$

$y = 0$

Simplify $x = 6 + 0$ .

Simplify the left side.

Remove parentheses.

$x = 6 + 0$

$y = 0$

$x = 6 + 0$

$y = 0$

Simplify the right side.

Add $6$ and $0$ .

$x = 6$

$y = 0$

$x = 6$

$y = 0$

$x = 6$

$y = 0$

$x = 6$

$y = 0$

Step 5

Replace all occurrences of $y$ with $4$ in each equation.

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

$x = 6 + 4$

$y = 4$

Simplify $x = 6 + 4$ .

Simplify the left side.

Remove parentheses.

$x = 6 + 4$

$y = 4$

$x = 6 + 4$

$y = 4$

Simplify the right side.

Add $6$ and $4$ .

$x = 10$

$y = 4$

$x = 10$

$y = 4$

$x = 10$

$y = 4$

$x = 10$

$y = 4$

Step 6

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

$(6, 0)$

$(10, 4)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(6, 0), (10, 4)$

Equation Form:

$x = 6, y = 0$

$x = 10, y = 4$

Step 8

Solve the System of Equations x^2-4y^2=36 x-y=6

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