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Solve the system of equations. $\newline$ $y = x^2 - 31x - 1$ $\newline$ $y = -29x + 34$ $\newline$ 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 = x^2 - 31x - 1

\newline

y = -29x + 34

\newline

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

\newline

(______,______)

\newline

(______,______)

Set Equations Equal: We have the system of equations: $\newline$ $y = x^2 - 31x - 1$ $\newline$ $y = -29x + 34$ $\newline$ To find the intersection points, set the two equations equal to each other. $\newline$ $x^2 - 31x - 1 = -29x + 34$
Form Quadratic Equation: Bring all terms to one side to form a quadratic equation. $\newline$ $x^2 - 31x - 1 + 29x - 34 = 0$ $\newline$ $x^2 - 2x - 35 = 0$
Factor Quadratic Equation: Factor the quadratic equation. $\newline$ We look for two numbers that multiply to $-35$ and add up to $-2$ . These numbers are $-7$ and $5$ . $\newline$ $x^2 - 7x + 5x - 35 = 0$ $\newline$ $(x - 7)(x + 5) = 0$
Solve for x: Solve for x. $\newline$ Set each factor equal to zero and solve for x. $\newline$ $(x - 7) = 0$ or $(x + 5) = 0$ $\newline$ $x = 7$ or $x = -5$
Find y-Values: Find the corresponding y-values for each x-value by substituting back into either of the original equations. We'll use $y = -29x + 34$ . $\newline$ For $x = 7$ : $\newline$ $y = -29(7) + 34$ $\newline$ $y = -203 + 34$ $\newline$ $y = -169$ $\newline$ For $x = -5$ : $\newline$ $y = -29(-5) + 34$ $\newline$ $y = 145 + 34$ $\newline$ $y = 179$
Write Coordinates: Write the coordinates in exact form. $\newline$ The intersection points are $(7, -169)$ and $(-5, 179)$ .

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