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

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Solve a system of linear and quadratic equations: circles

Full solution

Q. Solve the system of equations.

\newline

x^2 + y^2 = 2

\newline

x = y + 2

\newline

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

\newline

(______,______)

Substitute $x$ into first equation: Substitute $x$ from the second equation into the first equation. $\newline$ $x = y + 2$ $\newline$ $x^2 + y^2 = 2$ $\newline$ $(y + 2)^2 + y^2 = 2$
Expand and simplify: Expand and simplify the equation. $\newline$ $(y + 2)^2 + y^2 = 2$ $\newline$ $y^2 + 4y + 4 + y^2 = 2$ $\newline$ $2y^2 + 4y + 4 = 2$
Set equation to zero: Subtract $2$ from both sides to set the equation to zero. $\newline$ $2y^2 + 4y + 4 - 2 = 0$ $\newline$ $2y^2 + 4y + 2 = 0$
Divide and simplify: Divide the entire equation by $2$ to simplify. $\newline$ $y^2 + 2y + 1 = 0$
Factor the quadratic equation: Factor the quadratic equation. $\newline$ $(y + 1)(y + 1) = 0$
Solve for y: Solve for y by setting the factor equal to zero. $\newline$ $y + 1 = 0$ $\newline$ $y = -1$
Substitute $y$ into second equation: Substitute $y$ back into the second equation to find $x$ .
$x = y + 2$
$x = -1 + 2$
$x = 1$
Write coordinates: Write the coordinates in exact form. $\newline$ The coordinates are $(1, -1)$ .

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\newline

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\newline

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\newline

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Solve the system of equations.

\newline

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\newline

(\_,\_)

\newline

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\newline

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