Solve the system of equations. y = 37x^2 - 31x - 13 y = -31x + 24 Write the coordinates in exact form. Simplify all fractions and radicals. (______,______) (______,______)

Set Equations Equal: Set the two equations equal to each other to find the x-values where they intersect. 37x^2 - 31x - 13 = -31x + 24Form Quadratic Equation: Move all terms to one side to form a quadratic equation. 37x^2 - 31x - 13 + 31x - 24 = 0 37x^2 - 37 = 0Simplify and Factor: Simplify the equation by combining like terms. 37x^2 - 37 = 0 37(x^2 - 1) = 0Solve for x: Factor the quadratic equation. 37(x - 1)(x + 1) = 0Find y-Values: Solve for x by setting each factor equal to zero. (x - 1) = 0 or (x + 1) = 0 x = 1 or x = -1Write Coordinates: Find the corresponding y-values by substituting the x-values into one of the original equations. Let's use y = -31x + 24. For x = 1: y = -31(1) + 24 = -31 + 24 = -7 For x = -1: y = -31(-1) + 24 = 31 + 24 = 55Write the coordinates in exact form. First Coordinate: (1, -7) Second Coordinate: (-1, 55)

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Solve the system of equations. $\newline$ $y = 37x^2 - 31x - 13$ $\newline$ $y = -31x + 24$ $\newline$ Write the coordinates in exact form. Simplify all fractions and radicals. $\newline$ (,) $\newline$ (,)

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Algebra 1

Solve a system of linear and quadratic equations

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Q. Solve the system of equations.

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y = 37x^2 - 31x - 13

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y = -31x + 24

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

\newline

(______,______)

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(______,______)

Set Equations Equal: Set the two equations equal to each other to find the $x$ -values where they intersect. $37x^2 - 31x - 13 = -31x + 24$
Form Quadratic Equation: Move all terms to one side to form a quadratic equation. $\newline$ $37x^2 - 31x - 13 + 31x - 24 = 0$ $\newline$ $37x^2 - 37 = 0$
Simplify and Factor: Simplify the equation by combining like terms. $\newline$ $37x^2 - 37 = 0$ $\newline$ $37(x^2 - 1) = 0$
Solve for $x$ : Factor the quadratic equation. $37(x - 1)(x + 1) = 0$
Find y-Values: Solve for $x$ by setting each factor equal to zero. $\newline$ $(x - 1) = 0$ or $(x + 1) = 0$ $\newline$ $x = 1$ or $x = -1$
Write Coordinates: Find the corresponding $y$ -values by substituting the $x$ -values into one of the original equations. Let's use $y = -31x + 24$ .
For $x = 1$ : $y = -31(1) + 24 = -31 + 24 = -7$
For $x = -1$ : $y = -31(-1) + 24 = 31 + 24 = 55$
Write Coordinates: Find the corresponding $y$ -values by substituting the $x$ -values into one of the original equations. Let's use $y = -31x + 24$ . $\newline$ For $x = 1$ : $y = -31(1) + 24 = -31 + 24 = -7$ $\newline$ For $x = -1$ : $y = -31(-1) + 24 = 31 + 24 = 55$ Write the coordinates in exact form. $\newline$ First Coordinate: $(1, -7)$ $\newline$ Second Coordinate: $(-1, 55)$