Solve by Substitution x^2+2y^2=36 , x+y=6

$x^{2} + 2 y^{2} = 36$ , $x + y = 6$

Step 1

Subtract $y$ from both sides of the equation.

$x = 6 - y$

$x^{2} + 2 y^{2} = 36$

Step 2

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

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

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

$x = 6 - y$

Simplify the left side.

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

Simplify each term.

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

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

$x = 6 - y$

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

Apply the distributive property.

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

$x = 6 - y$

Apply the distributive property.

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

$x = 6 - y$

Apply the distributive property.

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

$x = 6 - y$

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

$x = 6 - y$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

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

$x = 6 - y$

Multiply $- 1$ by $6$ .

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

$x = 6 - y$

Multiply $6$ by $- 1$ .

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

$x = 6 - y$

Rewrite using the commutative property of multiplication.

$36 - 6 y - 6 y - 1 \cdot (- 1 y \cdot y) + 2 y^{2} = 36$

$x = 6 - y$

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

Move $y$ .

$36 - 6 y - 6 y - 1 \cdot (- 1 (y \cdot y)) + 2 y^{2} = 36$

$x = 6 - y$

Multiply $y$ by $y$ .

$36 - 6 y - 6 y - 1 \cdot (- 1 y^{2}) + 2 y^{2} = 36$

$x = 6 - y$

$36 - 6 y - 6 y - 1 \cdot (- 1 y^{2}) + 2 y^{2} = 36$

$x = 6 - y$

Multiply $- 1$ by $- 1$ .

$36 - 6 y - 6 y + 1 y^{2} + 2 y^{2} = 36$

$x = 6 - y$

Multiply $y^{2}$ by $1$ .

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

$x = 6 - y$

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

$x = 6 - y$

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

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

$x = 6 - y$

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

$x = 6 - y$

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

$x = 6 - y$

Add $y^{2}$ and $2 y^{2}$ .

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

$x = 6 - y$

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

$x = 6 - y$

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

$x = 6 - y$

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

$x = 6 - y$

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 $- 12 y + 3 y^{2}$ .

Reorder $- 12 y$ and $3 y^{2}$ .

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

$x = 6 - y$

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

$- 3 y (- y) - 12 y = 0$

$x = 6 - y$

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

$- 3 y (- y) - 3 y \cdot 4 = 0$

$x = 6 - y$

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

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

$x = 6 - y$

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

$x = 6 - y$

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

$y = 0$

$- y + 4 = 0$

$x = 6 - y$

Set $y$ equal to $0$ .

$y = 0$

$x = 6 - y$

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

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

$- y + 4 = 0$

$x = 6 - y$

Solve $- y + 4 = 0$ 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 $- 3 y (- y + 4) = 0$ 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 the right side.

Subtract $0$ from $6$ .

$x = 6$

$y = 0$

$x = 6$

$y = 0$

$x = 6$

$y = 0$

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 the right side.

Simplify $6 - (4)$ .

Multiply $- 1$ by $4$ .

$x = 6 - 4$

$y = 4$

Subtract $4$ from $6$ .

$x = 2$

$y = 4$

$x = 2$

$y = 4$

$x = 2$

$y = 4$

$x = 2$

$y = 4$

Step 6

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

$(6, 0)$

$(2, 4)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(6, 0), (2, 4)$

Equation Form:

$x = 6, y = 0$

$x = 2, y = 4$

Step 8

Solve by Substitution x^2+2y^2=36 , x+y=6

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