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Solve the system of equations. $\newline$ $y = x^2 + 12x - 41$ $\newline$ $y = 17x + 9$ $\newline$ 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

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Q. Solve the system of equations.

\newline

y = x^2 + 12x - 41

\newline

y = 17x + 9

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = x^2 + 12x - 41$ $\newline$ $y = 17x + 9$ $\newline$ To find the intersection points, set the two equations equal to each other. $\newline$ $x^2 + 12x - 41 = 17x + 9$
Rearrange and Solve: Rearrange the equation to bring all terms to one side and set it equal to zero. $\newline$ $x^2 + 12x - 41 - 17x - 9 = 0$ $\newline$ $x^2 - 5x - 50 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ We need to find two numbers that multiply to $-50$ and add up to $-5$ . These numbers are $-10$ and $5$ . $\newline$ $x^2 - 5x - 50 = (x - 10)(x + 5)$
Solve for x: Solve for x by setting each factor equal to zero. $\newline$ $(x - 10) = 0$ or $(x + 5) = 0$ $\newline$ $x = 10$ or $x = -5$
Substitute and Calculate: Find the corresponding $y$ -values for each $x$ -value by substituting back into either of the original equations. We'll use $y = 17x + 9$ . For $x = 10$ : $y = 17(10) + 9$ $y = 170 + 9$ $y = 179$
Find $y$ for $x=-5$ : Find the $y$ -value for $x = -5$ :
$y = 17(-5) + 9$
$y = -85 + 9$
$y = -76$
Write Coordinates: Write the coordinates in exact form. $\newline$ The first coordinate is $(10, 179)$ . $\newline$ The second coordinate is $(-5, -76)$ .

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