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Solve the system of equations. $\newline$ $y = -x - 12$ $\newline$ $x^2 + y^2 = 80$ $\newline$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,) $\newline$

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Solve a system of linear and quadratic equations: circles

Full solution

Q. Solve the system of equations.

\newline

y = -x - 12

\newline

x^2 + y^2 = 80

\newline

\newline

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

\newline

(______,______)

\newline

(______,______)

\newline

Substitute $y$ into second equation: Substitute $y$ from the first equation into the second equation. $\newline$ $y = -x - 12$ $\newline$ $x^2 + y^2 = 80$ $\newline$ $x^2 + (-x - 12)^2 = 80$
Expand and simplify: Expand and simplify the equation. $\newline$ $x^2 + (x^2 + 24x + 144) = 80$ $\newline$ $2x^2 + 24x + 144 = 80$
Subtract $80$ from both sides: Subtract $80$ from both sides to set the equation to zero. $\newline$ $2x^2 + 24x + 144 - 80 = 0$ $\newline$ $2x^2 + 24x + 64 = 0$
Divide and simplify: Divide the entire equation by $2$ to simplify. $\newline$ $x^2 + 12x + 32 = 0$
Factor the quadratic equation: Factor the quadratic equation. $\newline$ $(x + 8)(x + 4) = 0$
Set factors equal to zero: Set each factor equal to zero and solve for $x$ . $x + 8 = 0$ or $x + 4 = 0$ $x = -8$ or $x = -4$
Find corresponding y-values: Find the corresponding y-values for each x-value. $\newline$ For $x = -8$ : $y = -(-8) - 12 = 8 - 12 = -4$ $\newline$ For $x = -4$ : $y = -(-4) - 12 = 4 - 12 = -8$
Write coordinates in exact form: Write the coordinates in exact form. $\newline$ First Coordinate: $(-8, -4)$ $\newline$ Second Coordinate: $(-4, -8)$

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