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Solve the system of equations. $\newline$ $y = -36x + 26$ $\newline$ $y = x^2 - 46x - 13$ $\newline$ 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

Full solution

Q. Solve the system of equations.

\newline

y = -36x + 26

\newline

y = x^2 - 46x - 13

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = -36x + 26$ $\newline$ $y = x^2 - 46x - 13$ $\newline$ To find the intersection points, set the two equations equal to each other. $\newline$ $-36x + 26 = x^2 - 46x - 13$
Rearrange and Identify Quadratic: Rearrange the equation to set it to zero and identify the quadratic equation. $x^2 - 46x - 13 + 36x - 26 = 0$ $x^2 - 10x - 39 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ We need to find two numbers that multiply to $-39$ and add up to $-10$ . These numbers are $-13$ and $+3$ . $\newline$ $x^2 - 13x + 3x - 39 = 0$ $\newline$ $(x - 13)(x + 3) = 0$
Solve for x: Solve for x. $\newline$ Set each factor equal to zero and solve for x. $\newline$ $(x - 13) = 0$ or $(x + 3) = 0$ $\newline$ $x = 13$ or $x = -3$
Find Corresponding y-Values: Find the corresponding y-values for each x-value by substituting back into one of the original equations. $\newline$ For $x = 13$ : $\newline$ $y = -36(13) + 26$ $\newline$ $y = -468 + 26$ $\newline$ $y = -442$ $\newline$ For $x = -3$ : $\newline$ $y = -36(-3) + 26$ $\newline$ $y = 108 + 26$ $\newline$ $y = 134$
Write Coordinates: Write the coordinates in exact form. $\newline$ The intersection points are: $\newline$ First Coordinate: $(13, -442)$ $\newline$ Second Coordinate: $(-3, 134)$

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