Solve the system of equations. x = y - 13 x^2 + y^2 = 97 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Substitute x into second equation: Substitute x from the first equation into the second equation. x = y - 13 x^2 + y^2 = 97 (y - 13)^2 + y^2 = 97Expand and simplify: Expand the squared term and simplify the equation. (y - 13)^2 + y^2 = 97 y^2 - 26y + 169 + y^2 = 97 2y^2 - 26y + 169 = 97Set equation to zero: Subtract 97 from both sides to set the equation to zero. 2y^2 - 26y + 169 - 97 = 0 2y^2 - 26y + 72 = 0Divide and simplify: Divide the entire equation by 2 to simplify. y^2 - 13y + 36 = 0Factor the quadratic equation: Factor the quadratic equation. (y - 9)(y - 4) = 0Solve for y: Solve for y by setting each factor equal to zero. y - 9 = 0 or y - 4 = 0 y = 9 or y = 4Substitute y back into x: Substitute y values back into x = y - 13 to find corresponding x values. For y = 9: x = 9 - 13 = -4 For y = 4: x = 4 - 13 = -9

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Solve the system of equations. $\newline$ $x = y - 13$ $\newline$ $x^2 + y^2 = 97$ $\newline$ $\newline$ 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.

\newline

x = y - 13

\newline

x^2 + y^2 = 97

\newline

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

\newline

(______,______)

\newline

(______,______)

Substitute $x$ into second equation: Substitute $x$ from the first equation into the second equation. $\newline$ $x = y - 13$ $\newline$ $x^2 + y^2 = 97$ $\newline$ $(y - 13)^2 + y^2 = 97$
Expand and simplify: Expand the squared term and simplify the equation. $\newline$ $(y - 13)^2 + y^2 = 97$ $\newline$ $y^2 - 26y + 169 + y^2 = 97$ $\newline$ $2y^2 - 26y + 169 = 97$
Set equation to zero: Subtract $97$ from both sides to set the equation to zero. $\newline$ $2y^2 - 26y + 169 - 97 = 0$ $\newline$ $2y^2 - 26y + 72 = 0$
Divide and simplify: Divide the entire equation by $2$ to simplify. $\newline$ $y^2 - 13y + 36 = 0$
Factor the quadratic equation: Factor the quadratic equation. $\newline$ $(y - 9)(y - 4) = 0$
Solve for y: Solve for y by setting each factor equal to zero. $\newline$ $y - 9 = 0$ or $y - 4 = 0$ $\newline$ $y = 9$ or $y = 4$
Substitute $y$ back into $x$ : Substitute $y$ values back into $x = y - 13$ to find corresponding $x$ values. $\newline$ For $y = 9$ : $x = 9 - 13 = -4$ $\newline$ For $y = 4$ : $x = 4 - 13 = -9$