Solve the system of equations. y = -2x + 10 x^2 + y^2 = 65 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Substitute and Simplify: Substitute y from the first equation into the second equation. y = -2x + 10 x^2 + y^2 = 65 x^2 + (-2x + 10)^2 = 65Expand and Subtract: Expand and simplify the equation. x^2 + (-2x + 10)^2 = 65 x^2 + (4x^2 - 40x + 100) = 65 5x^2 - 40x + 100 = 65Divide and Simplify: Subtract 65 from both sides to set the equation to zero. 5x^2 - 40x + 100 - 65 = 0 5x^2 - 40x + 35 = 0Factor the Equation: Divide the entire equation by 5 to simplify. (5x^2 - 40x + 35) / 5 = 0 / 5 x^2 - 8x + 7 = 0Solve for x: Factor the quadratic equation. (x - 7)(x - 1) = 0Find y Values: Solve for x by setting each factor equal to zero. x - 7 = 0 or x - 1 = 0 x = 7 or x = 1Write Coordinates: Find the corresponding y values using the first equation y = -2x + 10. For x = 7: y = -2(7) + 10 = -14 + 10 = -4 For x = 1: y = -2(1) + 10 = -2 + 10 = 8Write the coordinates in exact form. First Coordinate: (7, -4) Second Coordinate: (1, 8)

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Solve the system of equations. $\newline$ $y = -2x + 10$ $\newline$ $x^2 + y^2 = 65$ $\newline$ 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 = -2x + 10

\newline

x^2 + y^2 = 65

\newline

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

\newline

(______,______)

\newline

(______,______)

Substitute and Simplify: Substitute $y$ from the first equation into the second equation. $\newline$ $y = -2x + 10$ $\newline$ $x^2 + y^2 = 65$ $\newline$ $x^2 + (-2x + 10)^2 = 65$
Expand and Subtract: Expand and simplify the equation. $\newline$ $x^2 + (-2x + 10)^2 = 65$ $\newline$ $x^2 + (4x^2 - 40x + 100) = 65$ $\newline$ $5x^2 - 40x + 100 = 65$
Divide and Simplify: Subtract $65$ from both sides to set the equation to zero. $\newline$ $5x^2 - 40x + 100 - 65 = 0$ $\newline$ $5x^2 - 40x + 35 = 0$
Factor the Equation: Divide the entire equation by $5$ to simplify. $\newline$ $\frac{5x^2 - 40x + 35}{5} = \frac{0}{5}$ $\newline$ $x^2 - 8x + 7 = 0$
Solve for $x$ : Factor the quadratic equation. $(x - 7)(x - 1) = 0$
Find y Values: Solve for $x$ by setting each factor equal to zero. $\newline$ $x - 7 = 0$ or $x - 1 = 0$ $\newline$ $x = 7$ or $x = 1$
Write Coordinates: Find the corresponding $y$ values using the first equation $y = -2x + 10$ . For $x = 7$ : $y = -2(7) + 10 = -14 + 10 = -4$ For $x = 1$ : $y = -2(1) + 10 = -2 + 10 = 8$
Write Coordinates: Find the corresponding $y$ values using the first equation $y = -2x + 10$ . For $x = 7$ : $y = -2(7) + 10 = -14 + 10 = -4$ For $x = 1$ : $y = -2(1) + 10 = -2 + 10 = 8$ Write the coordinates in exact form. First Coordinate: $(7, -4)$ Second Coordinate: $(1, 8)$