Solve the system of equations. x^2 + y^2 = 200 x = -3y - 20 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Substitute x into first equation: Substitute x from the second equation into the first equation. x^2 + y^2 = 200 x = -3y - 20 (-3y - 20)^2 + y^2 = 200Expand and simplify: Expand the squared term and simplify the equation. (-3y - 20)^2 + y^2 = 200 9y^2 + 120y + 400 + y^2 = 200 10y^2 + 120y + 400 = 200Set equation to zero: Subtract 200 from both sides to set the equation to zero. 10y^2 + 120y + 400 - 200 = 0 10y^2 + 120y + 200 = 0Divide and simplify: Divide the entire equation by 10 to simplify. 10y^2/10 + 120y/10 + 200/10 = 0/10 y^2 + 12y + 20 = 0Factor the equation: Factor the quadratic equation. (y + 10)(y + 2) = 0Solve for y: Solve for y by setting each factor equal to zero. y + 10 = 0 or y + 2 = 0 y = -10 or y = -2Find x values: Substitute y back into x = -3y - 20 to find the corresponding x values. For y = -10: x = -3(-10) - 20 x = 30 - 20 x = 10Find x values: Substitute y back into x = -3y - 20 to find the corresponding x values. For y = -2: x = -3(-2) - 20 x = 6 - 20 x = -14Write coordinates: Write the coordinates in exact form. First Coordinate: (10, -10) Second Coordinate: (-14, -2)

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Solve the system of equations. $\newline$ $x^2 + y^2 = 200$ $\newline$ $x = -3y - 20$ $\newline$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Algebra 2

Solve a system of linear and quadratic equations: circles

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Q. Solve the system of equations.

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x^2 + y^2 = 200

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x = -3y - 20

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Write the coordinates in exact form. Simplify all fractions and radicals.

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(______,______)

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(______,______)

Substitute $x$ into first equation: Substitute $x$ from the second equation into the first equation. $\newline$ $x^2 + y^2 = 200$ $\newline$ $x = -3y - 20$ $\newline$ $(-3y - 20)^2 + y^2 = 200$
Expand and simplify: Expand the squared term and simplify the equation. $\newline$ $(-3y - 20)^2 + y^2 = 200$ $\newline$ $9y^2 + 120y + 400 + y^2 = 200$ $\newline$ $10y^2 + 120y + 400 = 200$
Set equation to zero: Subtract $200$ from both sides to set the equation to zero. $\newline$ $10y^2 + 120y + 400 - 200 = 0$ $\newline$ $10y^2 + 120y + 200 = 0$
Divide and simplify: Divide the entire equation by $10$ to simplify. $\newline$ $\frac{10y^2}{10} + \frac{120y}{10} + \frac{200}{10} = \frac{0}{10}$ $\newline$ $y^2 + 12y + 20 = 0$
Factor the equation: Factor the quadratic equation. $\newline$ $(y + 10)(y + 2) = 0$
Solve for y: Solve for y by setting each factor equal to zero. $\newline$ $y + 10 = 0$ or $y + 2 = 0$ $\newline$ $y = -10$ or $y = -2$
Find $x$ values: Substitute $y$ back into $x = -3y - 20$ to find the corresponding $x$ values. $\newline$ For $y = -10$ : $x = -3(-10) - 20$ $\newline$ $x = 30 - 20$ $\newline$ $x = 10$
Find $x$ values: Substitute $y$ back into $x = -3y - 20$ to find the corresponding $x$ values. $\newline$ For $y = -2$ : $x = -3(-2) - 20$ $\newline$ $x = 6 - 20$ $\newline$ $x = -14$
Write coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(10, -10)$ $\newline$ Second Coordinate: $(-14, -2)$