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The graph shows the parabola y=−(x+4)2+5y=-(x+4)^2+5. Consider the following linear equation: y=−3x+by=-3x+b for some constant bb. If one of the solutions to the system of equations formed by the parabola and the linear equation is (−4,5)(-4,5), which of the following is the other solution?

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Q. The graph shows the parabola y=−(x+4)2+5y=-(x+4)^2+5. Consider the following linear equation: y=−3x+by=-3x+b for some constant bb. If one of the solutions to the system of equations formed by the parabola and the linear equation is (−4,5)(-4,5), which of the following is the other solution?
  1. Find bb in linear equation: First, we need to find the value of bb in the linear equation y=−3x+by=-3x+b using the given solution (−4,5)(-4,5).\newlineSubstitute xx with −4-4 and yy with 55 into the linear equation.\newline5=−3(−4)+b5 = -3(-4) + b\newline5=12+b5 = 12 + b\newlineNow, solve for bb.\newlinebb11\newlinebb22
  2. Set equations equal: Now that we have the linear equation y=−3x−7y=-3x-7, we can set it equal to the parabola equation y=−(x+4)2+5y=-(x+4)^2+5 to find the other point of intersection.−(x+4)2+5=−3x−7-(x+4)^2+5 = -3x-7
  3. Expand and simplify: Next, we need to expand the left side of the equation and simplify.\newline−x2−8x−16+5=−3x−7-x^2 - 8x - 16 + 5 = -3x - 7\newline−x2−8x−11=−3x−7-x^2 - 8x - 11 = -3x - 7
  4. Move terms and solve: Now, we will move all terms to one side to set the equation to zero and solve for xx.−x2−8x+3x−11+7=0-x^2 - 8x + 3x - 11 + 7 = 0−x2−5x−4=0-x^2 - 5x - 4 = 0
  5. Factor quadratic equation: We can solve the quadratic equation −x2−5x−4=0-x^2 - 5x - 4 = 0 by factoring or using the quadratic formula. However, since we already know one solution is x=−4x = -4, we can factor by grouping.\newlinex=−4x = -4 is a solution, so (x+4)(x + 4) is a factor.\newlineLet's factor −x2−5x−4-x^2 - 5x - 4.\newline−x2−4x−x−4=0-x^2 - 4x - x - 4 = 0\newline−x(x+4)−1(x+4)=0-x(x + 4) - 1(x + 4) = 0\newline(x+4)(−x−1)=0(x + 4)(-x - 1) = 0
  6. Find y-coordinate: The solutions to the equation (x+4)(−x−1)=0(x + 4)(-x - 1) = 0 are x=−4x = -4 and x=−1x = -1. We already know the solution (−4,5)(-4,5), so we need to find the y-coordinate when x=−1x = -1. Substitute x=−1x = -1 into the linear equation y=−3x−7y = -3x - 7. y=−3(−1)−7y = -3(-1) - 7 y=3−7y = 3 - 7 y=−4y = -4
  7. Final solution: The other solution to the system of equations is (−1,−4)(-1,-4).

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