The figure shows the graph of $y=3x+33$ . Which of the following is one of the two solutions $(x,y)$ to the system of equations formed by this line and the curve determined by $y =-3(x+10)^2+9$ ?

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Algebra 2

Find the vertex of the transformed function

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Q. The figure shows the graph of

y=3x+33

. Which of the following is one of the two solutions

(x,y)

to the system of equations formed by this line and the curve determined by

y =-3(x+10)^2+9

Identify Equations: : Identify the system of equations we need to solve. $\newline$ We have two equations: $\newline$ $1$ . $y = 3x + 33$ (linear equation) $\newline$ $2$ . $y = -3(x + 10)^2 + 9$ (quadratic equation) $\newline$ To find the intersection points, we need to set these two equations equal to each other and solve for $x$ .
Set Equations Equal: : Set the two equations equal to each other. $\newline$ $3x + 33 = -3(x + 10)^2 + 9$ $\newline$ This will allow us to find the $x$ -values where the line and the curve intersect.
Expand and Simplify: : Expand the quadratic equation and simplify. $\newline$ $-3(x + 10)^2 + 9 = -3(x^2 + 20x + 100) + 9$ $\newline$ $= -3x^2 - 60x - 300 + 9$ $\newline$ Now we have: $\newline$ $3x + 33 = -3x^2 - 60x - 300 + 9$
Combine and Move Terms: : Combine like terms and move all terms to one side to set the equation to zero. $\newline$ $0 = -3x^2 - 60x - 300 + 9 - 3x - 33$ $\newline$ $0 = -3x^2 - 63x - 324$
Simplify Equation: : Simplify the equation by dividing all terms by $-3$ to make it easier to solve. $\newline$ $0 = x^2 + 21x + 108$
Factor Quadratic: : Factor the quadratic equation. $\newline$ $(x + 9)(x + 12) = 0$
Solve for x: : Solve for x by setting each factor equal to zero. $\newline$ $x + 9 = 0$ or $x + 12 = 0$ $\newline$ $x = -9$ or $x = -12$
Substitute and Find $y$ : : Substitute the $x$ -values back into one of the original equations to find the corresponding $y$ -values. $\newline$ For $x = -9$ : $\newline$ $y = 3(-9) + 33 = -27 + 33 = 6$ $\newline$ For $x = -12$ : $\newline$ $y = 3(-12) + 33 = -36 + 33 = -3$
Write Solutions: : Write down the solutions to the system of equations. $\newline$ The solutions are $(-9, 6)$ and $(-12, -3)$ .