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Solve the system of equations. $\newline$ $y = x - 18$ $\newline$ $x^2 + y^2 = 290$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Solve a system of linear and quadratic equations: circles

Full solution

Q. Solve the system of equations.

\newline

y = x - 18

\newline

x^2 + y^2 = 290

\newline

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

\newline

(______,______)

\newline

(______,______)

Substitute $y$ Equation: Substitute $y$ from the first equation into the second equation. $\newline$ $y = x - 18$ $\newline$ $x^2 + y^2 = 290$ $\newline$ $x^2 + (x - 18)^2 = 290$
Expand and Simplify: Expand the second term and simplify the equation. $\newline$ $x^2 + (x^2 - 36x + 324) = 290$ $\newline$ $2x^2 - 36x + 324 = 290$
Subtract to Zero: Subtract $290$ from both sides to set the equation to zero. $\newline$ $2x^2 - 36x + 324 - 290 = 0$ $\newline$ $2x^2 - 36x + 34 = 0$
Divide and Simplify: Divide the entire equation by $2$ to simplify. $\newline$ $x^2 - 18x + 17 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ $(x - 17)(x - 1) = 0$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $x - 17 = 0$ or $x - 1 = 0$ $\newline$ $x = 17$ or $x = 1$
Substitute $x$ Values: Substitute $x$ values back into the first equation to find $y$ values. $\newline$ For $x = 17$ : $y = 17 - 18 = -1$ $\newline$ For $x = 1$ : $y = 1 - 18 = -17$
Write Coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(17, -1)$ $\newline$ Second Coordinate: $(1, -17)$

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