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

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

Step 1

Eliminate the equal sides of each equation and combine.

$6 - x = x^{2} - 6 x + 6$

Step 2

Solve $6 - x = x^{2} - 6 x + 6$ for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} - 6 x + 6 = 6 - x$

Move all terms containing $x$ to the left side of the equation.

Add $x$ to both sides of the equation.

$x^{2} - 6 x + 6 + x = 6$

Add $- 6 x$ and $x$ .

$x^{2} - 5 x + 6 = 6$

Subtract $6$ from both sides of the equation.

$x^{2} - 5 x + 6 - 6 = 0$

Combine the opposite terms in $x^{2} - 5 x + 6 - 6$ .

Subtract $6$ from $6$ .

$x^{2} - 5 x + 0 = 0$

Add $x^{2} - 5 x$ and $0$ .

$x^{2} - 5 x = 0$

Factor $x$ out of $x^{2} - 5 x$ .

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

$x \cdot x - 5 x = 0$

Factor $x$ out of $- 5 x$ .

$x \cdot x + x \cdot - 5 = 0$

Factor $x$ out of $x \cdot x + x \cdot - 5$ .

$x (x - 5) = 0$

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

$x = 0$

$x - 5 = 0 + y = x^{2} - 6 x + 6$

Set $x$ equal to $0$ .

$x = 0$

Set $x - 5$ equal to $0$ and solve for $x$ .

Set $x - 5$ equal to $0$ .

$x - 5 = 0$

Add $5$ to both sides of the equation.

$x = 5$

The final solution is all the values that make $x (x - 5) = 0$ true.

$x = 0, 5$

Step 3

Evaluate $y$ when $x = 0$ .

Substitute $0$ for $x$ .

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

Substitute $0$ for $x$ in $y = {(0)}^{2} - 6 \cdot 0 + 6$ and solve for $y$ .

Remove parentheses.

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

Remove parentheses.

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

Simplify ${(0)}^{2} - 6 \cdot 0 + 6$ .

Simplify each term.

Raising $0$ to any positive power yields $0$ .

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

Multiply $- 6$ by $0$ .

$y = 0 + 0 + 6$

Simplify by adding numbers.

Add $0$ and $0$ .

$y = 0 + 6$

Add $0$ and $6$ .

$y = 6$

Step 4

Evaluate $y$ when $x = 5$ .

Substitute $5$ for $x$ .

$y = {(5)}^{2} - 6 \cdot 5 + 6$

Substitute $5$ for $x$ in $y = {(5)}^{2} - 6 \cdot 5 + 6$ and solve for $y$ .

Remove parentheses.

$y = 5^{2} - 6 \cdot 5 + 6$

Remove parentheses.

$y = {(5)}^{2} - 6 \cdot 5 + 6$

Simplify ${(5)}^{2} - 6 \cdot 5 + 6$ .

Simplify each term.

Raise $5$ to the power of $2$ .

$y = 25 - 6 \cdot 5 + 6$

Multiply $- 6$ by $5$ .

$y = 25 - 30 + 6$

Simplify by adding and subtracting.

Subtract $30$ from $25$ .

$y = - 5 + 6$

Add $- 5$ and $6$ .

$y = 1$

Step 5

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

$(0, 6)$

$(5, 1)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(0, 6), (5, 1)$

Equation Form:

$x = 0, y = 6$

$x = 5, y = 1$

Step 7

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

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