Precalculus Examples

$x^{2} + 2 y^{2} = 441$ , $x + y = 21$

Step 1

Subtract $y$ from both sides of the equation.

$x = 21 - y$

$x^{2} + 2 y^{2} = 441$

Step 2

Replace all occurrences of $x$ with $21 - y$ in each equation.

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $x^{2} + 2 y^{2} = 441$ with $21 - y$ .

${(21 - y)}^{2} + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

Simplify ${(21 - y)}^{2} + 2 y^{2}$ .

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

Rewrite ${(21 - y)}^{2}$ as $(21 - y) (21 - y)$ .

$(21 - y) (21 - y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.2

Expand $(21 - y) (21 - y)$ using the FOIL Method.

Tap for more steps...

Step 2.2.1.1.2.1

Apply the distributive property.

$21 (21 - y) - y (21 - y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.2.2

Apply the distributive property.

$21 \cdot 21 + 21 (- y) - y (21 - y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.2.3

Apply the distributive property.

$21 \cdot 21 + 21 (- y) - y \cdot 21 - y (- y) + 2 y^{2} = 441$

$x = 21 - y$

$21 \cdot 21 + 21 (- y) - y \cdot 21 - y (- y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3

Simplify and combine like terms.

Tap for more steps...

Step 2.2.1.1.3.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.3.1.1

Multiply $21$ by $21$ .

$441 + 21 (- y) - y \cdot 21 - y (- y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.1.2

Multiply $- 1$ by $21$ .

$441 - 21 y - y \cdot 21 - y (- y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.1.3

Multiply $21$ by $- 1$ .

$441 - 21 y - 21 y - y (- y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.1.4

Rewrite using the commutative property of multiplication.

$441 - 21 y - 21 y - 1 \cdot (- 1 y \cdot y) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.1.5

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

Tap for more steps...

Step 2.2.1.1.3.1.5.1

Move $y$ .

$441 - 21 y - 21 y - 1 \cdot (- 1 (y \cdot y)) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.1.5.2

Multiply $y$ by $y$ .

$441 - 21 y - 21 y - 1 \cdot (- 1 y^{2}) + 2 y^{2} = 441$

$x = 21 - y$

$441 - 21 y - 21 y - 1 \cdot (- 1 y^{2}) + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.1.6

Multiply $- 1$ by $- 1$ .

$441 - 21 y - 21 y + 1 y^{2} + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.1.7

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

$441 - 21 y - 21 y + y^{2} + 2 y^{2} = 441$

$x = 21 - y$

$441 - 21 y - 21 y + y^{2} + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.1.3.2

Subtract $21 y$ from $- 21 y$ .

$441 - 42 y + y^{2} + 2 y^{2} = 441$

$x = 21 - y$

$441 - 42 y + y^{2} + 2 y^{2} = 441$

$x = 21 - y$

$441 - 42 y + y^{2} + 2 y^{2} = 441$

$x = 21 - y$

Step 2.2.1.2

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

$441 - 42 y + 3 y^{2} = 441$

$x = 21 - y$

$441 - 42 y + 3 y^{2} = 441$

$x = 21 - y$

$441 - 42 y + 3 y^{2} = 441$

$x = 21 - y$

$441 - 42 y + 3 y^{2} = 441$

$x = 21 - y$

Step 3

Solve for $y$ in $441 - 42 y + 3 y^{2} = 441$ .

Tap for more steps...

Step 3.1

Subtract $441$ from both sides of the equation.

$441 - 42 y + 3 y^{2} - 441 = 0$

$x = 21 - y$

Step 3.2

Combine the opposite terms in $441 - 42 y + 3 y^{2} - 441$ .

Tap for more steps...

Step 3.2.1

Subtract $441$ from $441$ .

$- 42 y + 3 y^{2} + 0 = 0$

$x = 21 - y$

Step 3.2.2

Add $- 42 y + 3 y^{2}$ and $0$ .

$- 42 y + 3 y^{2} = 0$

$x = 21 - y$

$- 42 y + 3 y^{2} = 0$

$x = 21 - y$

Step 3.3

Factor $- 3 y$ out of $- 42 y + 3 y^{2}$ .

Tap for more steps...

Step 3.3.1

Reorder $- 42 y$ and $3 y^{2}$ .

$3 y^{2} - 42 y = 0$

$x = 21 - y$

Step 3.3.2

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

$- 3 y (- y) - 42 y = 0$

$x = 21 - y$

Step 3.3.3

Factor $- 3 y$ out of $- 42 y$ .

$- 3 y (- y) - 3 y \cdot 14 = 0$

$x = 21 - y$

Step 3.3.4

Factor $- 3 y$ out of $- 3 y (- y) - 3 y (14)$ .

$- 3 y (- y + 14) = 0$

$x = 21 - y$

$- 3 y (- y + 14) = 0$

$x = 21 - y$

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 $0$ .

$y = 0$

$- y + 14 = 0$

$x = 21 - y$

Step 3.5

Set $y$ equal to $0$ .

$y = 0$

$x = 21 - y$

Step 3.6

Set $- y + 14$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 3.6.1

Set $- y + 14$ equal to $0$ .

$- y + 14 = 0$

$x = 21 - y$

Step 3.6.2

Solve $- y + 14 = 0$ for $y$ .

Tap for more steps...

Step 3.6.2.1

Subtract $14$ from both sides of the equation.

$- y = - 14$

$x = 21 - y$

Step 3.6.2.2

Divide each term in $- y = - 14$ by $- 1$ and simplify.

Tap for more steps...

Step 3.6.2.2.1

Divide each term in $- y = - 14$ by $- 1$ .

$\frac{- y}{- 1} = \frac{- 14}{- 1}$

$x = 21 - y$

Step 3.6.2.2.2

Simplify the left side.

Tap for more steps...

Step 3.6.2.2.2.1

Dividing two negative values results in a positive value.

$\frac{y}{1} = \frac{- 14}{- 1}$

$x = 21 - y$

Step 3.6.2.2.2.2

Divide $y$ by $1$ .

$y = \frac{- 14}{- 1}$

$x = 21 - y$

$y = \frac{- 14}{- 1}$

$x = 21 - y$

Step 3.6.2.2.3

Simplify the right side.

Tap for more steps...

Step 3.6.2.2.3.1

Divide $- 14$ by $- 1$ .

$y = 14$

$x = 21 - y$

$y = 14$

$x = 21 - y$

$y = 14$

$x = 21 - y$

$y = 14$

$x = 21 - y$

$y = 14$

$x = 21 - y$

Step 3.7

The final solution is all the values that make $- 3 y (- y + 14) = 0$ true.

$y = 0, 14$

$x = 21 - y$

$y = 0, 14$

$x = 21 - y$

Step 4

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

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $x = 21 - y$ with $0$ .

$x = 21 - (0)$

$y = 0$

Step 4.2

Simplify the right side.

Tap for more steps...

Step 4.2.1

Subtract $0$ from $21$ .

$x = 21$

$y = 0$

$x = 21$

$y = 0$

$x = 21$

$y = 0$

Step 5

Replace all occurrences of $y$ with $14$ in each equation.

Tap for more steps...

Step 5.1

Replace all occurrences of $y$ in $x = 21 - y$ with $14$ .

$x = 21 - (14)$

$y = 14$

Step 5.2

Simplify the right side.

Tap for more steps...

Step 5.2.1

Simplify $21 - (14)$ .

Tap for more steps...

Step 5.2.1.1

Multiply $- 1$ by $14$ .

$x = 21 - 14$

$y = 14$

Step 5.2.1.2

Subtract $14$ from $21$ .

$x = 7$

$y = 14$

$x = 7$

$y = 14$

$x = 7$

$y = 14$

$x = 7$

$y = 14$

Step 6

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

$(21, 0)$

$(7, 14)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(21, 0), (7, 14)$

Equation Form:

$x = 21, y = 0$

$x = 7, y = 14$

Step 8