Precalculus Examples

$x^{2} + y^{2} = 68$ , $x - y = 6$

Step 1

Add $y$ to both sides of the equation.

$x = 6 + y$

$x^{2} + y^{2} = 68$

Step 2

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

Tap for more steps...

Step 2.1

Replace all occurrences of $x$ in $x^{2} + y^{2} = 68$ with $6 + y$ .

${(6 + y)}^{2} + y^{2} = 68$

$x = 6 + y$

Step 2.2

Simplify the left side.

Tap for more steps...

Step 2.2.1

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

Tap for more steps...

Step 2.2.1.1

Simplify each term.

Tap for more steps...

Step 2.2.1.1.1

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

$(6 + y) (6 + y) + y^{2} = 68$

$x = 6 + y$

Step 2.2.1.1.2

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

Tap for more steps...

Step 2.2.1.1.2.1

Apply the distributive property.

$6 (6 + y) + y (6 + y) + y^{2} = 68$

$x = 6 + y$

Step 2.2.1.1.2.2

Apply the distributive property.

$6 \cdot 6 + 6 y + y (6 + y) + y^{2} = 68$

$x = 6 + y$

Step 2.2.1.1.2.3

Apply the distributive property.

$6 \cdot 6 + 6 y + y \cdot 6 + y \cdot y + y^{2} = 68$

$x = 6 + y$

$6 \cdot 6 + 6 y + y \cdot 6 + y \cdot y + y^{2} = 68$

$x = 6 + y$

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 $6$ by $6$ .

$36 + 6 y + y \cdot 6 + y \cdot y + y^{2} = 68$

$x = 6 + y$

Step 2.2.1.1.3.1.2

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

$36 + 6 y + 6 \cdot y + y \cdot y + y^{2} = 68$

$x = 6 + y$

Step 2.2.1.1.3.1.3

Multiply $y$ by $y$ .

$36 + 6 y + 6 y + y^{2} + y^{2} = 68$

$x = 6 + y$

$36 + 6 y + 6 y + y^{2} + y^{2} = 68$

$x = 6 + y$

Step 2.2.1.1.3.2

Add $6 y$ and $6 y$ .

$36 + 12 y + y^{2} + y^{2} = 68$

$x = 6 + y$

$36 + 12 y + y^{2} + y^{2} = 68$

$x = 6 + y$

$36 + 12 y + y^{2} + y^{2} = 68$

$x = 6 + y$

Step 2.2.1.2

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

$36 + 12 y + 2 y^{2} = 68$

$x = 6 + y$

$36 + 12 y + 2 y^{2} = 68$

$x = 6 + y$

$36 + 12 y + 2 y^{2} = 68$

$x = 6 + y$

$36 + 12 y + 2 y^{2} = 68$

$x = 6 + y$

Step 3

Solve for $y$ in $36 + 12 y + 2 y^{2} = 68$ .

Tap for more steps...

Step 3.1

Subtract $68$ from both sides of the equation.

$36 + 12 y + 2 y^{2} - 68 = 0$

$x = 6 + y$

Step 3.2

Subtract $68$ from $36$ .

$12 y + 2 y^{2} - 32 = 0$

$x = 6 + y$

Step 3.3

Factor the left side of the equation.

Tap for more steps...

Step 3.3.1

Factor $2$ out of $12 y + 2 y^{2} - 32$ .

Tap for more steps...

Step 3.3.1.1

Factor $2$ out of $12 y$ .

$2 (6 y) + 2 y^{2} - 32 = 0$

$x = 6 + y$

Step 3.3.1.2

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

$2 (6 y) + 2 (y^{2}) - 32 = 0$

$x = 6 + y$

Step 3.3.1.3

Factor $2$ out of $- 32$ .

$2 (6 y) + 2 y^{2} + 2 \cdot - 16 = 0$

$x = 6 + y$

Step 3.3.1.4

Factor $2$ out of $2 (6 y) + 2 y^{2}$ .

$2 (6 y + y^{2}) + 2 \cdot - 16 = 0$

$x = 6 + y$

Step 3.3.1.5

Factor $2$ out of $2 (6 y + y^{2}) + 2 \cdot - 16$ .

$2 (6 y + y^{2} - 16) = 0$

$x = 6 + y$

$2 (6 y + y^{2} - 16) = 0$

$x = 6 + y$

Step 3.3.2

Let $u = y$ . Substitute $u$ for all occurrences of $y$ .

$2 (6 u + u^{2} - 16) = 0$

$x = 6 + y$

Step 3.3.3

Factor $6 u + u^{2} - 16$ using the AC method.

Tap for more steps...

Step 3.3.3.1

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 $- 16$ and whose sum is $6$ .

$- 2, 8$

$x = 6 + y$

Step 3.3.3.2

Write the factored form using these integers.

$2 ((u - 2) (u + 8)) = 0$

$x = 6 + y$

$2 ((u - 2) (u + 8)) = 0$

$x = 6 + y$

Step 3.3.4

Factor.

Tap for more steps...

Step 3.3.4.1

Replace all occurrences of $u$ with $y$ .

$2 ((y - 2) (y + 8)) = 0$

$x = 6 + y$

Step 3.3.4.2

Remove unnecessary parentheses.

$2 (y - 2) (y + 8) = 0$

$x = 6 + y$

$2 (y - 2) (y + 8) = 0$

$x = 6 + y$

$2 (y - 2) (y + 8) = 0$

$x = 6 + y$

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 $0$ .

$y - 2 = 0$

$y + 8 = 0$

$x = 6 + y$

Step 3.5

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

Tap for more steps...

Step 3.5.1

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

$y - 2 = 0$

$x = 6 + y$

Step 3.5.2

Add $2$ to both sides of the equation.

$y = 2$

$x = 6 + y$

$y = 2$

$x = 6 + y$

Step 3.6

Set $y + 8$ equal to $0$ and solve for $y$ .

Tap for more steps...

Step 3.6.1

Set $y + 8$ equal to $0$ .

$y + 8 = 0$

$x = 6 + y$

Step 3.6.2

Subtract $8$ from both sides of the equation.

$y = - 8$

$x = 6 + y$

$y = - 8$

$x = 6 + y$

Step 3.7

The final solution is all the values that make $2 (y - 2) (y + 8) = 0$ true.

$y = 2, - 8$

$x = 6 + y$

$y = 2, - 8$

$x = 6 + y$

Step 4

Replace all occurrences of $y$ with $2$ in each equation.

Tap for more steps...

Step 4.1

Replace all occurrences of $y$ in $x = 6 + y$ with $2$ .

$x = 6 + 2$

$y = 2$

Step 4.2

Simplify $x = 6 + 2$ .

Tap for more steps...

Step 4.2.1

Simplify the left side.

Tap for more steps...

Step 4.2.1.1

Remove parentheses.

$x = 6 + 2$

$y = 2$

$x = 6 + 2$

$y = 2$

Step 4.2.2

Simplify the right side.

Tap for more steps...

Step 4.2.2.1

Add $6$ and $2$ .

$x = 8$

$y = 2$

$x = 8$

$y = 2$

$x = 8$

$y = 2$

$x = 8$

$y = 2$

Step 5

Replace all occurrences of $y$ with $- 8$ in each equation.

Tap for more steps...

Step 5.1

Replace all occurrences of $y$ in $x = 6 + y$ with $- 8$ .

$x = 6 - 8$

$y = - 8$

Step 5.2

Simplify $x = 6 - 8$ .

Tap for more steps...

Step 5.2.1

Simplify the left side.

Tap for more steps...

Step 5.2.1.1

Remove parentheses.

$x = 6 - 8$

$y = - 8$

$x = 6 - 8$

$y = - 8$

Step 5.2.2

Simplify the right side.

Tap for more steps...

Step 5.2.2.1

Subtract $8$ from $6$ .

$x = - 2$

$y = - 8$

$x = - 2$

$y = - 8$

$x = - 2$

$y = - 8$

$x = - 2$

$y = - 8$

Step 6

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

$(8, 2)$

$(- 2, - 8)$

Step 7

The result can be shown in multiple forms.

Point Form:

$(8, 2), (- 2, - 8)$

Equation Form:

$x = 8, y = 2$

$x = - 2, y = - 8$

Step 8