Solve by Substitution x^2-y=0 , x+y=6

$x^{2} - y = 0$ , $x + y = 6$

Step 1

Subtract $y$ from both sides of the equation.

$x = 6 - y$

$x^{2} - y = 0$

Step 2

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

Replace all occurrences of $x$ in $x^{2} - y = 0$ with $6 - y$ .

${(6 - y)}^{2} - y = 0$

$x = 6 - y$

Simplify the left side.

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

Simplify each term.

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

$(6 - y) (6 - y) - y = 0$

$x = 6 - y$

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

Apply the distributive property.

$6 (6 - y) - y (6 - y) - y = 0$

$x = 6 - y$

Apply the distributive property.

$6 \cdot 6 + 6 (- y) - y (6 - y) - y = 0$

$x = 6 - y$

Apply the distributive property.

$6 \cdot 6 + 6 (- y) - y \cdot 6 - y (- y) - y = 0$

$x = 6 - y$

$6 \cdot 6 + 6 (- y) - y \cdot 6 - y (- y) - y = 0$

$x = 6 - y$

Simplify and combine like terms.

Simplify each term.

Multiply $6$ by $6$ .

$36 + 6 (- y) - y \cdot 6 - y (- y) - y = 0$

$x = 6 - y$

Multiply $- 1$ by $6$ .

$36 - 6 y - y \cdot 6 - y (- y) - y = 0$

$x = 6 - y$

Multiply $6$ by $- 1$ .

$36 - 6 y - 6 y - y (- y) - y = 0$

$x = 6 - y$

Rewrite using the commutative property of multiplication.

$36 - 6 y - 6 y - 1 \cdot (- 1 y \cdot y) - y = 0$

$x = 6 - y$

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

Move $y$ .

$36 - 6 y - 6 y - 1 \cdot (- 1 (y \cdot y)) - y = 0$

$x = 6 - y$

Multiply $y$ by $y$ .

$36 - 6 y - 6 y - 1 \cdot (- 1 y^{2}) - y = 0$

$x = 6 - y$

$36 - 6 y - 6 y - 1 \cdot (- 1 y^{2}) - y = 0$

$x = 6 - y$

Multiply $- 1$ by $- 1$ .

$36 - 6 y - 6 y + 1 y^{2} - y = 0$

$x = 6 - y$

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

$36 - 6 y - 6 y + y^{2} - y = 0$

$x = 6 - y$

$36 - 6 y - 6 y + y^{2} - y = 0$

$x = 6 - y$

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

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

$x = 6 - y$

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

$x = 6 - y$

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

$x = 6 - y$

Subtract $y$ from $- 12 y$ .

$36 + y^{2} - 13 y = 0$

$x = 6 - y$

$36 + y^{2} - 13 y = 0$

$x = 6 - y$

$36 + y^{2} - 13 y = 0$

$x = 6 - y$

$36 + y^{2} - 13 y = 0$

$x = 6 - y$

Step 3

Solve for $y$ in $36 + y^{2} - 13 y = 0$ .

Factor $36 + y^{2} - 13 y$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $36$ and whose sum is $- 13$ .

$- 9, - 4$

$x = 6 - y$

Write the factored form using these integers.

$(y - 9) (y - 4) = 0$

$x = 6 - y$

$(y - 9) (y - 4) = 0$

$x = 6 - y$

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

$y - 9 = 0$

$y - 4 = 0$

$x = 6 - y$

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

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

$y - 9 = 0$

$x = 6 - y$

Add $9$ to both sides of the equation.

$y = 9$

$x = 6 - y$

$y = 9$

$x = 6 - y$

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

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

$y - 4 = 0$

$x = 6 - y$

Add $4$ to both sides of the equation.

$y = 4$

$x = 6 - y$

$y = 4$

$x = 6 - y$

The final solution is all the values that make $(y - 9) (y - 4) = 0$ true.

$y = 9, 4$

$x = 6 - y$

$y = 9, 4$

$x = 6 - y$

Step 4

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

Replace all occurrences of $y$ in $x = 6 - y$ with $9$ .

$x = 6 - (9)$

$y = 9$

Simplify the right side.

Simplify $6 - (9)$ .

Multiply $- 1$ by $9$ .

$x = 6 - 9$

$y = 9$

Subtract $9$ from $6$ .

$x = - 3$

$y = 9$

$x = - 3$

$y = 9$

$x = - 3$

$y = 9$

$x = - 3$

$y = 9$

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.

$(- 3, 9)$

$(2, 4)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(- 3, 9), (2, 4)$

Equation Form:

$x = - 3, y = 9$

$x = 2, y = 4$

Step 8

Solve by Substitution x^2-y=0 , x+y=6

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