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{:[y=x^(2)+7x-7],[y=x+9]:} If (x,y) is a solution to the system of equations and x < 0, what is the value of y ?

y=x2+7x−7y=x+9 \begin{array}{l} y=x^{2}+7 x-7 \\ y=x+9 \end{array} \newlineIf (x,y) (x, y) is a solution to the system of equations and x<0 , what is the value of y y ?

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Q. y=x2+7x−7y=x+9 \begin{array}{l} y=x^{2}+7 x-7 \\ y=x+9 \end{array} \newlineIf (x,y) (x, y) is a solution to the system of equations and x<0 x<0 , what is the value of y y ?
  1. Write equations: Write down the system of equations.\newlineWe have the following system of equations:\newliney=x2+7x−7y = x^2 + 7x - 7\newliney=x+9y = x + 9
  2. Set equations equal: Set the two equations equal to each other to find the xx-values where their yy-values are the same.\newlinex2+7x−7=x+9x^2 + 7x - 7 = x + 9
  3. Subtract xx: Subtract xx from both sides to start solving for xx.\newlinex2+7x−x−7=9x^2 + 7x - x - 7 = 9\newlinex2+6x−7=9x^2 + 6x - 7 = 9
  4. Subtract 99: Subtract 99 from both sides to set the equation to zero.x2+6x−7−9=0x^2 + 6x - 7 - 9 = 0x2+6x−16=0x^2 + 6x - 16 = 0
  5. Factor quadratic equation: Factor the quadratic equation.\newlineWe are looking for two numbers that multiply to −16-16 and add up to 66. These numbers are 88 and −2-2.\newline(x+8)(x−2)=0(x + 8)(x - 2) = 0
  6. Solve for x: Solve for x.\newlineSet each factor equal to zero and solve for x.\newlinex+8=0x + 8 = 0 or x−2=0x - 2 = 0\newlinex=−8x = -8 or x=2x = 2
  7. Use x=−8x = -8: Since we are given that x < 0, we will use x=−8x = -8 and discard x=2x = 2.
  8. Substitute x=−8x = -8: Substitute x=−8x = -8 into either of the original equations to find the corresponding value of yy.\newlineUsing the second equation y=x+9y = x + 9:\newliney=(−8)+9y = (-8) + 9\newliney=1y = 1

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