The parabola shown is defined by the equation y=(1)/(3)x^(2)-2x-4. If the line defined by the equation y=-(8)/(3)(x+5)+9 is also graphed, at what point (x,y) will the line and parabola intersect?" Choose 1 answer: (A) (0,-4) (B) (4,-(20)/(3)) (c) (3,-7) (D) (-1,-(5)/(3)) quations | Lesson quations - Basic example quations - Harder example

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The parabola shown is defined by the equation  y=(1)/(3)x^(2)-2x-4. If the line defined by the equation  y=-(8)/(3)(x+5)+9 is also graphed, at what point  (x,y) will the line and parabola intersect?" Choose 1 answer: (A)  (0,-4) (B)  (4,-(20)/(3)) (c)  (3,-7) (D)  (-1,-(5)/(3)) quations | Lesson quations - Basic example quations - Harder example

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Set Equations Equal: To find the intersection point(s) of the line and the parabola, we need to set their equations equal to each other and solve for x. y = (1/3)x^2 - 2x - 4 (Parabola) y = -(8/3)(x + 5) + 9 (Line) (1/3)x^2 - 2x - 4 = -(8/3)(x + 5) + 9Simplify and Solve: Now we will simplify and solve the equation for x. (1/3)x^2 - 2x - 4 = -(8/3)x - (8/3)*5 + 9 (1/3)x^2 - 2x - 4 = -(8/3)x - 40/3 + 9 To clear the fractions, multiply every term by 3: x^2 - 6x - 12 = -8x - 40 + 27Combine Like Terms: Combine like terms and move all terms to one side to form a quadratic equation: x^2 - 6x - 12 + 8x + 13 = 0 x^2 + 2x + 1 = 0Factor Quadratic Equation: The quadratic equation x^2 + 2x + 1 can be factored as (x + 1)^2 = 0.Solve for x: Solve for x by taking the square root of both sides: x + 1 = 0 x = -1Substitute x and Solve for y: Now that we have the x-coordinate of the intersection point, we need to substitute x back into either the original line or parabola equation to find the corresponding y-coordinate. Let's substitute x into the line's equation: y = -(8/3)(-1 + 5) + 9Calculate y-coordinate: Calculate the y-coordinate: y = -(8/3)(4) + 9 y = -32/3 + 9 y = -32/3 + 27/3 y = -5/3Find Intersection Point: The intersection point (x,y) is (-1, -5/3), which corresponds to answer choice (D).

The parabola shown is defined by the equation $y=\frac{1}{3} x^{2}-2 x-4$ . If the line defined by the equation $y=-\frac{8}{3}(x+5)+9$ is also graphed, at what point $(x, y)$ will the line and parabola intersect?

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The parabola shown is defined by the equation $y=\frac{1}{3} x^{2}-2 x-4$ . If the line defined by the equation $y=-\frac{8}{3}(x+5)+9$ is also graphed, at what point $(x, y)$ will the line and parabola intersect?