Solve the System of Equations y=x+5 x^2+y^2=97

$y = x + 5$ $x^{2} + y^{2} = 97$

Step 1

Replace all occurrences of $y$ with $x + 5$ in each equation.

Replace all occurrences of $y$ in $x^{2} + y^{2} = 97$ with $x + 5$ .

$x^{2} + {(x + 5)}^{2} = 97$

$y = x + 5$

Simplify the left side.

Simplify $x^{2} + {(x + 5)}^{2}$ .

Simplify each term.

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$x^{2} + (x + 5) (x + 5) = 97$

$y = x + 5$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$x^{2} + x (x + 5) + 5 (x + 5) = 97$

$y = x + 5$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 5 + 5 (x + 5) = 97$

$y = x + 5$

Apply the distributive property.

$x^{2} + x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 = 97$

$y = x + 5$

$x^{2} + x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5 = 97$

$y = x + 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$x^{2} + x^{2} + x \cdot 5 + 5 x + 5 \cdot 5 = 97$

$y = x + 5$

Move $5$ to the left of $x$ .

$x^{2} + x^{2} + 5 \cdot x + 5 x + 5 \cdot 5 = 97$

$y = x + 5$

Multiply $5$ by $5$ .

$x^{2} + x^{2} + 5 x + 5 x + 25 = 97$

$y = x + 5$

$x^{2} + x^{2} + 5 x + 5 x + 25 = 97$

$y = x + 5$

Add $5 x$ and $5 x$ .

$x^{2} + x^{2} + 10 x + 25 = 97$

$y = x + 5$

$x^{2} + x^{2} + 10 x + 25 = 97$

$y = x + 5$

$x^{2} + x^{2} + 10 x + 25 = 97$

$y = x + 5$

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

$2 x^{2} + 10 x + 25 = 97$

$y = x + 5$

$2 x^{2} + 10 x + 25 = 97$

$y = x + 5$

$2 x^{2} + 10 x + 25 = 97$

$y = x + 5$

$2 x^{2} + 10 x + 25 = 97$

$y = x + 5$

Step 2

Solve for $x$ in $2 x^{2} + 10 x + 25 = 97$ .

Subtract $97$ from both sides of the equation.

$2 x^{2} + 10 x + 25 - 97 = 0$

$y = x + 5$

Subtract $97$ from $25$ .

$2 x^{2} + 10 x - 72 = 0$

$y = x + 5$

Factor the left side of the equation.

Factor $2$ out of $2 x^{2} + 10 x - 72$ .

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

$2 (x^{2}) + 10 x - 72 = 0$

$y = x + 5$

Factor $2$ out of $10 x$ .

$2 (x^{2}) + 2 (5 x) - 72 = 0$

$y = x + 5$

Factor $2$ out of $- 72$ .

$2 x^{2} + 2 (5 x) + 2 \cdot - 36 = 0$

$y = x + 5$

Factor $2$ out of $2 x^{2} + 2 (5 x)$ .

$2 (x^{2} + 5 x) + 2 \cdot - 36 = 0$

$y = x + 5$

Factor $2$ out of $2 (x^{2} + 5 x) + 2 \cdot - 36$ .

$2 (x^{2} + 5 x - 36) = 0$

$y = x + 5$

$2 (x^{2} + 5 x - 36) = 0$

$y = x + 5$

Factor.

Factor $x^{2} + 5 x - 36$ 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 $5$ .

$- 4, 9$

$y = x + 5$

Write the factored form using these integers.

$2 ((x - 4) (x + 9)) = 0$

$y = x + 5$

$2 ((x - 4) (x + 9)) = 0$

$y = x + 5$

Remove unnecessary parentheses.

$2 (x - 4) (x + 9) = 0$

$y = x + 5$

$2 (x - 4) (x + 9) = 0$

$y = x + 5$

$2 (x - 4) (x + 9) = 0$

$y = x + 5$

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

$x - 4 = 0$

$x + 9 = 0$

$y = x + 5$

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

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

$x - 4 = 0$

$y = x + 5$

Add $4$ to both sides of the equation.

$x = 4$

$y = x + 5$

$x = 4$

$y = x + 5$

Set $x + 9$ equal to $0$ and solve for $x$ .

Set $x + 9$ equal to $0$ .

$x + 9 = 0$

$y = x + 5$

Subtract $9$ from both sides of the equation.

$x = - 9$

$y = x + 5$

$x = - 9$

$y = x + 5$

The final solution is all the values that make $2 (x - 4) (x + 9) = 0$ true.

$x = 4, - 9$

$y = x + 5$

$x = 4, - 9$

$y = x + 5$

Step 3

Replace all occurrences of $x$ with $4$ in each equation.

Replace all occurrences of $x$ in $y = x + 5$ with $4$ .

$y = (4) + 5$

$x = 4$

Simplify the right side.

Add $4$ and $5$ .

$y = 9$

$x = 4$

$y = 9$

$x = 4$

$y = 9$

$x = 4$

Step 4

Replace all occurrences of $x$ with $- 9$ in each equation.

Replace all occurrences of $x$ in $y = x + 5$ with $- 9$ .

$y = (- 9) + 5$

$x = - 9$

Simplify the right side.

Add $- 9$ and $5$ .

$y = - 4$

$x = - 9$

$y = - 4$

$x = - 9$

$y = - 4$

$x = - 9$

Step 5

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

$(4, 9)$

$(- 9, - 4)$

Step 6

The result can be shown in multiple forms.

Point Form:

$(4, 9), (- 9, - 4)$

Equation Form:

$x = 4, y = 9$

$x = - 9, y = - 4$

Step 7

Solve the System of Equations y=x+5 x^2+y^2=97

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