Solve the system of equations. $\newline$ $y = -x - 24$ $\newline$ $x^2 + y^2 = 488$ $\newline$ 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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y = -x - 24

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x^2 + y^2 = 488

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

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(\_,\_)

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(\_,\_)

Substitute and Simplify: Substitute the expression for $y$ from the first equation into the second equation. $\newline$ Given the system of equations: $\newline$ $y = -x - 24$ $\newline$ $x^2 + y^2 = 488$ $\newline$ We substitute $y$ in the second equation with the expression from the first equation: $\newline$ $x^2 + (-x - 24)^2 = 488$
Expand and Combine Terms: Expand the squared term and simplify the equation. $\newline$ $x^2 + (-x - 24)^2 = 488$ $\newline$ $x^2 + (x^2 + 48x + 576) = 488$ $\newline$ Combine like terms: $\newline$ $2x^2 + 48x + 576 = 488$
Move and Simplify Further: Move all terms to one side to set the equation to zero and simplify further. $\newline$ $2x^2 + 48x + 576 - 488 = 0$ $\newline$ $2x^2 + 48x + 88 = 0$ $\newline$ Divide the entire equation by $2$ to simplify: $\newline$ $x^2 + 24x + 44 = 0$
Factor the Quadratic: Factor the quadratic equation. $\newline$ We look for two numbers that multiply to $44$ and add up to $24$ . These numbers are $22$ and $2$ . $\newline$ $(x + 22)(x + 2) = 0$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $x + 22 = 0$ or $x + 2 = 0$ $\newline$ $x = -22$ or $x = -2$
Find y-Values: Find the corresponding y-values for each x-value by substituting back into the first equation. $\newline$ For $x = -22$ : $\newline$ $y = -(-22) - 24$ $\newline$ $y = 22 - 24$ $\newline$ $y = -2$ $\newline$ For $x = -2$ : $\newline$ $y = -(-2) - 24$ $\newline$ $y = 2 - 24$ $\newline$ $y = -22$
Write Coordinates: Write the coordinates in exact form. $\newline$ The solutions to the system of equations are the points where the values of $x$ and $y$ satisfy both equations. Therefore, the coordinates are: $\newline$ $(-22, -2)$ and $(-2, -22)$