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Solve the system of equations. $\newline$ $x = 3y - 10$ $\newline$ $x^2 + y^2 = 50$ $\newline$ 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

x = 3y - 10

\newline

x^2 + y^2 = 50

\newline

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

\newline

(______,______)

\newline

(______,______)

Substitute $x$ : Substitute $x$ from the first equation into the second equation. $\newline$ $x = 3y - 10$ $\newline$ $x^2 + y^2 = 50$ $\newline$ $(3y - 10)^2 + y^2 = 50$
Expand and simplify: Expand and simplify the equation. $\newline$ $(3y - 10)^2 + y^2 = 50$ $\newline$ $9y^2 - 60y + 100 + y^2 = 50$ $\newline$ $10y^2 - 60y + 100 = 50$
Subtract $50$ : Subtract $50$ from both sides to set the equation to zero. $\newline$ $10y^2 - 60y + 100 - 50 = 0$ $\newline$ $10y^2 - 60y + 50 = 0$
Divide and simplify: Divide the entire equation by $10$ to simplify. $\newline$ $y^2 - 6y + 5 = 0$
Factor quadratic equation: Factor the quadratic equation. $\newline$ $(y - 5)(y - 1) = 0$
Solve for y: Solve for y by setting each factor equal to zero. $\newline$ $y - 5 = 0$ or $y - 1 = 0$ $\newline$ $y = 5$ or $y = 1$
Substitute $y$ into first equation: Substitute $y$ back into the first equation to find $x$ . $\newline$ For $y = 5$ : $x = 3(5) - 10 = 15 - 10 = 5$ $\newline$ For $y = 1$ : $x = 3(1) - 10 = 3 - 10 = -7$
Write coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(5, 5)$ $\newline$ Second Coordinate: $(-7, 1)$

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