Solve the system of equations. y = -x + 13 x^2 + y^2 = 109 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Substitute y into second equation: Substitute y from the first equation into the second equation. y = -x + 13 x^2 + y^2 = 109 x^2 + (-x + 13)^2 = 109Expand and simplify: Expand and simplify the equation. x^2 + (-x + 13)^2 = 109 x^2 + (x^2 - 26x + 169) = 109 2x^2 - 26x + 169 = 109Move 109 to left side: Move 109 to the left side of the equation. 2x^2 - 26x + 169 - 109 = 0 2x^2 - 26x + 60 = 0Divide and simplify: Divide the equation by 2 to simplify. x^2 - 13x + 30 = 0Factor the quadratic equation: Factor the quadratic equation. (x - 10)(x - 3) = 0Solve for x: Solve for x by setting each factor equal to zero. x - 10 = 0 or x - 3 = 0 x = 10 or x = 3Find corresponding y values: Find the corresponding y values for each x. For x = 10: y = -10 + 13 = 3 For x = 3: y = -3 + 13 = 10Write coordinates in exact form: Write the coordinates in exact form. First Coordinate: (10, 3) Second Coordinate: (3, 10)

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Solve the system of equations. $\newline$ $y = -x + 13$ $\newline$ $x^2 + y^2 = 109$ $\newline$ $\newline$ 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

y = -x + 13

\newline

x^2 + y^2 = 109

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

\newline

(______,______)

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(______,______)

Substitute $y$ into second equation: Substitute $y$ from the first equation into the second equation. $\newline$ $y = -x + 13$ $\newline$ $x^2 + y^2 = 109$ $\newline$ $x^2 + (-x + 13)^2 = 109$
Expand and simplify: Expand and simplify the equation. $\newline$ $x^2 + (-x + 13)^2 = 109$ $\newline$ $x^2 + (x^2 - 26x + 169) = 109$ $\newline$ $2x^2 - 26x + 169 = 109$
Move $109$ to left side: Move $109$ to the left side of the equation. $\newline$ $2x^2 - 26x + 169 - 109 = 0$ $\newline$ $2x^2 - 26x + 60 = 0$
Divide and simplify: Divide the equation by $2$ to simplify. $\newline$ $x^2 - 13x + 30 = 0$
Factor the quadratic equation: Factor the quadratic equation. $\newline$ $(x - 10)(x - 3) = 0$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $x - 10 = 0$ or $x - 3 = 0$ $\newline$ $x = 10$ or $x = 3$
Find corresponding y values: Find the corresponding y values for each x. $\newline$ For $x = 10$ : $y = -10 + 13 = 3$ $\newline$ For $x = 3$ : $y = -3 + 13 = 10$
Write coordinates in exact form: Write the coordinates in exact form. $\newline$ First Coordinate: $(10, 3)$ $\newline$ Second Coordinate: $(3, 10)$