Solve the System of Equations y=x^2+7x-13 6=x-y

$y = x^{2} + 7 x - 13$ $6 = x - y$

Step 1

Replace all occurrences of $y$ with $x^{2} + 7 x - 13$ in each equation.

Replace all occurrences of $y$ in $6 = x - y$ with $x^{2} + 7 x - 13$ .

$6 = x - (x^{2} + 7 x - 13)$

$y = x^{2} + 7 x - 13$

Simplify the right side.

Simplify $x - (x^{2} + 7 x - 13)$ .

Simplify each term.

Apply the distributive property.

$6 = x - x^{2} - (7 x) + 13$

$y = x^{2} + 7 x - 13$

Simplify.

Multiply $7$ by $- 1$ .

$6 = x - x^{2} - 7 x + 13$

$y = x^{2} + 7 x - 13$

Multiply $- 1$ by $- 13$ .

$6 = x - x^{2} - 7 x + 13$

$y = x^{2} + 7 x - 13$

$6 = x - x^{2} - 7 x + 13$

$y = x^{2} + 7 x - 13$

$6 = x - x^{2} - 7 x + 13$

$y = x^{2} + 7 x - 13$

Subtract $7 x$ from $x$ .

$6 = - x^{2} - 6 x + 13$

$y = x^{2} + 7 x - 13$

$6 = - x^{2} - 6 x + 13$

$y = x^{2} + 7 x - 13$

$6 = - x^{2} - 6 x + 13$

$y = x^{2} + 7 x - 13$

$6 = - x^{2} - 6 x + 13$

$y = x^{2} + 7 x - 13$

Step 2

Solve for $x$ in $6 = - x^{2} - 6 x + 13$ .

Rewrite the equation as $- x^{2} - 6 x + 13 = 6$ .

$- x^{2} - 6 x + 13 = 6$

$y = x^{2} + 7 x - 13$

Subtract $6$ from both sides of the equation.

$- x^{2} - 6 x + 13 - 6 = 0$

$y = x^{2} + 7 x - 13$

Subtract $6$ from $13$ .

$- x^{2} - 6 x + 7 = 0$

$y = x^{2} + 7 x - 13$

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - 6 x + 7$ .

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 6 x + 7 = 0$

$y = x^{2} + 7 x - 13$

Factor $- 1$ out of $- 6 x$ .

$- (x^{2}) - (6 x) + 7 = 0$

$y = x^{2} + 7 x - 13$

Rewrite $7$ as $- 1 (- 7)$ .

$- (x^{2}) - (6 x) - 1 \cdot - 7 = 0$

$y = x^{2} + 7 x - 13$

Factor $- 1$ out of $- (x^{2}) - (6 x)$ .

$- (x^{2} + 6 x) - 1 \cdot - 7 = 0$

$y = x^{2} + 7 x - 13$

Factor $- 1$ out of $- (x^{2} + 6 x) - 1 (- 7)$ .

$- (x^{2} + 6 x - 7) = 0$

$y = x^{2} + 7 x - 13$

$- (x^{2} + 6 x - 7) = 0$

$y = x^{2} + 7 x - 13$

Factor.

Factor $x^{2} + 6 x - 7$ 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 $- 7$ and whose sum is $6$ .

$- 1, 7$

$y = x^{2} + 7 x - 13$

Write the factored form using these integers.

$- ((x - 1) (x + 7)) = 0$

$y = x^{2} + 7 x - 13$

$- ((x - 1) (x + 7)) = 0$

$y = x^{2} + 7 x - 13$

Remove unnecessary parentheses.

$- (x - 1) (x + 7) = 0$

$y = x^{2} + 7 x - 13$

$- (x - 1) (x + 7) = 0$

$y = x^{2} + 7 x - 13$

$- (x - 1) (x + 7) = 0$

$y = x^{2} + 7 x - 13$

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

$x - 1 = 0$

$x + 7 = 0$

$y = x^{2} + 7 x - 13$

Set $x - 1$ equal to $0$ and solve for $x$ .

Set $x - 1$ equal to $0$ .

$x - 1 = 0$

$y = x^{2} + 7 x - 13$

Add $1$ to both sides of the equation.

$x = 1$

$y = x^{2} + 7 x - 13$

$x = 1$

$y = x^{2} + 7 x - 13$

Set $x + 7$ equal to $0$ and solve for $x$ .

Set $x + 7$ equal to $0$ .

$x + 7 = 0$

$y = x^{2} + 7 x - 13$

Subtract $7$ from both sides of the equation.

$x = - 7$

$y = x^{2} + 7 x - 13$

$x = - 7$

$y = x^{2} + 7 x - 13$

The final solution is all the values that make $- (x - 1) (x + 7) = 0$ true.

$x = 1, - 7$

$y = x^{2} + 7 x - 13$

$x = 1, - 7$

$y = x^{2} + 7 x - 13$

Step 3

Replace all occurrences of $x$ with $1$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} + 7 x - 13$ with $1$ .

$y = {(1)}^{2} + 7 (1) - 13$

$x = 1$

Simplify the right side.

Simplify ${(1)}^{2} + 7 (1) - 13$ .

Simplify each term.

One to any power is one.

$y = 1 + 7 (1) - 13$

$x = 1$

Multiply $7$ by $1$ .

$y = 1 + 7 - 13$

$x = 1$

$y = 1 + 7 - 13$

$x = 1$

Simplify by adding and subtracting.

Add $1$ and $7$ .

$y = 8 - 13$

$x = 1$

Subtract $13$ from $8$ .

$y = - 5$

$x = 1$

$y = - 5$

$x = 1$

$y = - 5$

$x = 1$

$y = - 5$

$x = 1$

$y = - 5$

$x = 1$

Step 4

Replace all occurrences of $x$ with $- 7$ in each equation.

Replace all occurrences of $x$ in $y = x^{2} + 7 x - 13$ with $- 7$ .

$y = {(- 7)}^{2} + 7 (- 7) - 13$

$x = - 7$

Simplify the right side.

Simplify ${(- 7)}^{2} + 7 (- 7) - 13$ .

Simplify each term.

Raise $- 7$ to the power of $2$ .

$y = 49 + 7 (- 7) - 13$

$x = - 7$

Multiply $7$ by $- 7$ .

$y = 49 - 49 - 13$

$x = - 7$

$y = 49 - 49 - 13$

$x = - 7$

Simplify by subtracting numbers.

Subtract $49$ from $49$ .

$y = 0 - 13$

$x = - 7$

Subtract $13$ from $0$ .

$y = - 13$

$x = - 7$

$y = - 13$

$x = - 7$

$y = - 13$

$x = - 7$

$y = - 13$

$x = - 7$

$y = - 13$

$x = - 7$

Step 5

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

$(1, - 5)$

$(- 7, - 13)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(1, - 5), (- 7, - 13)$

Equation Form:

$x = 1, y = - 5$

$x = - 7, y = - 13$

Step 7

Solve the System of Equations y=x^2+7x-13 6=x-y

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