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Solve the system of equations. $\newline$ $y = 2x - 15$ $\newline$ $x^2 + y^2 = 50$ $\newline$ 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

y = 2x - 15

\newline

x^2 + y^2 = 50

\newline

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

\newline

(______,______)

\newline

(______,______)

Substitute $y$ : Substitute $y$ from the first equation into the second equation. $\newline$ $y = 2x - 15$ $\newline$ $x^2 + y^2 = 50$ $\newline$ $x^2 + (2x - 15)^2 = 50$
Expand and simplify: Expand the squared term and simplify the equation. $\newline$ $x^2 + (2x - 15)(2x - 15) = 50$ $\newline$ $x^2 + 4x^2 - 60x + 225 = 50$ $\newline$ $5x^2 - 60x + 225 = 50$
Subtract to set to zero: Subtract $50$ from both sides to set the equation to zero. $\newline$ $5x^2 - 60x + 225 - 50 = 0$ $\newline$ $5x^2 - 60x + 175 = 0$
Divide to simplify: Divide the entire equation by $5$ to simplify. $\newline$ $x^2 - 12x + 35 = 0$
Factor the equation: Factor the quadratic equation. $\newline$ $(x - 7)(x - 5) = 0$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $x - 7 = 0$ or $x - 5 = 0$ $\newline$ $x = 7$ or $x = 5$
Find corresponding y values: Find the corresponding y values using the first equation. $\newline$ For $x = 7$ : $y = 2(7) - 15 = 14 - 15 = -1$ $\newline$ For $x = 5$ : $y = 2(5) - 15 = 10 - 15 = -5$
Write coordinates: Write the coordinates in exact form. $\newline$ First Coordinate: $(7, -1)$ $\newline$ Second Coordinate: $(5, -5)$

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